Daily Papers

Foundation models for biology and physics optimize predictive accuracy, but their internal representations systematically fail to preserve the continuous geometry of the systems they model. We identify the root cause: the Geometric Alignment Tax, an intrinsic cost of forcing continuous manifolds through discrete categorical bottlenecks. Controlled ablations on synthetic dynamical systems demonstrate that replacing cross-entropy with a continuous head on an identical encoder reduces geometric distortion by up to 8.5x, while learned codebooks exhibit a non-monotonic double bind where finer quantization worsens geometry despite improving reconstruction. Under continuous objectives, three architectures differ by 1.3x; under discrete tokenization, they diverge by 3,000x. Evaluating 14 biological foundation models with rate-distortion theory and MINE, we identify three failure regimes: Local-Global Decoupling, Representational Compression, and Geometric Vacuity. A controlled experiment confirms that Evo 2's reverse-complement robustness on real DNA reflects conserved sequence composition, not learned symmetry. No model achieves simultaneously low distortion, high mutual information, and global coherence.

Analysis of learned representations has a blind spot: it focuses on similarity, measuring how closely embeddings align with external references, but similarity reveals only what is represented, not whether that structure is robust. We introduce geometric stability, a distinct dimension that quantifies how reliably representational geometry holds under perturbation, and present Shesha, a framework for measuring it. Across 2,463 configurations in seven domains, we show that stability and similarity are empirically uncorrelated (ρapprox 0.01) and mechanistically distinct: similarity metrics collapse after removing the top principal components, while stability retains sensitivity to fine-grained manifold structure. This distinction yields actionable insights: for safety monitoring, stability acts as a functional geometric canary, detecting structural drift nearly 2times more sensitively than CKA while filtering out the non-functional noise that triggers false alarms in rigid distance metrics; for controllability, supervised stability predicts linear steerability (ρ= 0.89-0.96); for model selection, stability dissociates from transferability, revealing a geometric tax that transfer optimization incurs. Beyond machine learning, stability predicts CRISPR perturbation coherence and neural-behavioral coupling. By quantifying how reliably systems maintain structure, geometric stability provides a necessary complement to similarity for auditing representations across biological and computational systems.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

We present a computational framework that transforms single images into 3D physical objects. The visual geometry of a physical object in an image is determined by three orthogonal attributes: mechanical properties, external forces, and rest-shape geometry. Existing single-view 3D reconstruction methods often overlook this underlying composition, presuming rigidity or neglecting external forces. Consequently, the reconstructed objects fail to withstand real-world physical forces, resulting in instability or undesirable deformation -- diverging from their intended designs as depicted in the image. Our optimization framework addresses this by embedding physical compatibility into the reconstruction process. We explicitly decompose the three physical attributes and link them through static equilibrium, which serves as a hard constraint, ensuring that the optimized physical shapes exhibit desired physical behaviors. Evaluations on a dataset collected from Objaverse demonstrate that our framework consistently enhances the physical realism of 3D models over existing methods. The utility of our framework extends to practical applications in dynamic simulations and 3D printing, where adherence to physical compatibility is paramount.

Vision-language models solve geometry problems with rising accuracy, yet their intermediate states remain latent and unverifiable: a relation expressed in textual reasoning or drawing code carries no guarantee that a constraint-satisfying configuration realizes it. We observe that existing externalization methods based on rendered pixels or one-shot scripts fail to provide exact, per-action geometric guarantees. Enforcing geometric relations by algebraic definition closes this gap: the workspace becomes a constraint-checked evolving canvas. We present Draw2Think, a framework that recasts geometric reasoning from latent spatial inference into agentic interaction with the GeoGebra constraint engine. In a Propose-Draw-Verify loop, Draw2Think externalizes hypotheses onto an executable canvas, measures exact geometric quantities, and feeds structured observations back to the model, so subsequent reasoning proceeds from checked canvas state grounded by the shared workspace. This externalization makes two properties separately auditable: model-level Construction Fidelity (whether the canvas realizes the intended configuration) and engine-level Measurement Faithfulness (exact values and relations from canvas constraints). Across construction, outcome, and rendering evaluations, Draw2Think builds canvases that pass 95.9% predicate-level and 84.0% strict problem-level construction checks on GeoGoal, improves outcome accuracy by up to 4.1%/16.4% on planar/solid benchmarks, and attains 68.2%/90.5% strict/relaxed rendering scores on GenExam-math. Project page is available at https://draw2think.github.io/

Since Isaac Newton first studied the Kissing Number Problem in 1694, determining the maximal number of non-overlapping spheres around a central sphere has remained a fundamental challenge. This problem is the local analogue of Hilbert's 18th problem, bridging geometry, number theory, and information theory. Although significant progress has been made through lattices and codes, the irregularities of high-dimensional geometry, dimensional structure variability, and combinatorial explosion beyond Go limit the scalability and generality of existing methods. Here we model the problem as a two-player matrix completion game and train the reinforcement learning system, PackingStar, to play the games. The matrix entries represent pairwise cosines of sphere center vectors. One player fills entries while another corrects suboptimal ones to improve exploration quality, cooperatively maximizing the matrix size, corresponding to the kissing number. These matrices are decomposed into representative substructures, providing diverse bases and structural constraints that steer subsequent games and make extremely large spaces tractable. PackingStar surpasses records from dimensions 25 to 31 and sets new lower bounds for generalized kissing numbers under various angular constraints. It achieves the first breakthrough beyond rational structures from 1971 in 13 dimensions and discovers over 6000 new structures in other dimensions. Notably, some configurations challenge long-held antipodal paradigms, revealing algebraic correspondences with finite simple groups as well as geometric relationships across dimensions. Inspired by these patterns, humans devised further improved constructions. These results demonstrate AI's power to explore high-dimensional spaces beyond human intuition via extreme-scale reinforcement learning and open new pathways for the Kissing Number Problem and broader geometry research.

Large vision-language models have significantly advanced GUI agents, enabling executable interaction across web, mobile, and desktop interfaces. Yet these gains largely rely on a forgiving region-tolerant paradigm, where many nearby pixels inside the same component remain valid. Precise geometric construction breaks this assumption: actions must land on points in continuous canvas space rather than tolerant regions. Because geometric primitives carry ontological dependencies, a local coordinate error can induce cascading topological failures that distort downstream objects and invalidate the final construction. We identify this regime as precision-sensitive GUI tasks, requiring point-level accuracy, geometry-aware verification, and robustness to dependency-driven error propagation. To benchmark it, we introduce PAGE Bench, with 4,906 problems and over 224K process-supervised, pixel-level GUI actions. We further propose PAGER, a topology-aware agent that decomposes construction into dependency-structured planning and pixel-level execution. Pixel-grounded supervised tuning establishes executable action grammar, while precision-aligned reinforcement learning mitigates rollout-induced exposure bias through state-conditioned geometric feedback. Experiments reveal a pronounced Semantic-Execution Gap: general multimodal models can exceed 88% action type accuracy yet remain below 6% task success. PAGER closes this gap, delivering 4.1x higher task success than the strongest evaluated general baseline and raising step success rate from below 9% for GUI-specialized agents to over 62%, establishing a new state of the art for point-precise GUI control.

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges, such as handling thin structures and non-watertight geometries, which limit their flexibility and accuracy. In contrast, we propose a novel geometric data representation that models geometry as distributions-a powerful representation that makes no assumptions about surface genus, connectivity, or boundary conditions. Our approach uses diffusion models with a novel network architecture to learn surface point distributions, capturing fine-grained geometric details. We evaluate our representation qualitatively and quantitatively across various object types, demonstrating its effectiveness in achieving high geometric fidelity. Additionally, we explore applications using our representation, such as textured mesh representation, neural surface compression, dynamic object modeling, and rendering, highlighting its potential to advance 3D geometric learning.

Deep sequence models are said to store atomic facts predominantly in the form of associative memory: a brute-force lookup of co-occurring entities. We identify a dramatically different form of storage of atomic facts that we term as geometric memory. Here, the model has synthesized embeddings encoding novel global relationships between all entities, including ones that do not co-occur in training. Such storage is powerful: for instance, we show how it transforms a hard reasoning task involving an ell-fold composition into an easy-to-learn 1-step navigation task. From this phenomenon, we extract fundamental aspects of neural embedding geometries that are hard to explain. We argue that the rise of such a geometry, as against a lookup of local associations, cannot be straightforwardly attributed to typical supervisory, architectural, or optimizational pressures. Counterintuitively, a geometry is learned even when it is more complex than the brute-force lookup. Then, by analyzing a connection to Node2Vec, we demonstrate how the geometry stems from a spectral bias that -- in contrast to prevailing theories -- indeed arises naturally despite the lack of various pressures. This analysis also points out to practitioners a visible headroom to make Transformer memory more strongly geometric. We hope the geometric view of parametric memory encourages revisiting the default intuitions that guide researchers in areas like knowledge acquisition, capacity, discovery, and unlearning.

In machine learning, there is a long history of trying to build neural networks that can learn from fewer example data by baking in strong geometric priors. However, it is not always clear a priori what geometric constraints are appropriate for a given task. Here, we explore the possibility that one can uncover useful geometric inductive biases by studying how training molds the Riemannian geometry induced by unconstrained neural network feature maps. We first show that at infinite width, neural networks with random parameters induce highly symmetric metrics on input space. This symmetry is broken by feature learning: networks trained to perform classification tasks learn to magnify local areas along decision boundaries. This holds in deep networks trained on high-dimensional image classification tasks, and even in self-supervised representation learning. These results begin to elucidate how training shapes the geometry induced by unconstrained neural network feature maps, laying the groundwork for an understanding of this richly nonlinear form of feature learning.

Neural representations carry rich geometric structure; but does that structure causally shape behavior? To address this question, we intervene along paths through activation space defined by different geometries, and measure the behavioral trajectories they induce. In particular, we test whether interventions that respect the geometry of activation space will yield behaviors close to those the model exhibits naturally. Concretely, we first fit an activation manifold M_h to representations and a behavior manifold M_y to output probability distributions. We then test the link M_h leftrightarrow M_y via interventions: we find that steering along M_h, which we term manifold steering, yields behavioral trajectories that follow M_y, while linear steering -- which assumes a Euclidean geometry -- cuts through off-manifold regions and hence produces unnatural outputs. Moreover, optimizing interventions in activation space to produce paths along M_y recovers activation trajectories that trace the curvature of M_h. We demonstrate this bidirectional relationship between the geometry of representation and behavior across tasks and modalities. In language models, we use reasoning tasks with cyclic and sequential geometries as well as in-context learning tasks with more complex graph geometries. In a video world model, we use a task with geometry corresponding to physical dynamics. Overall, our work shows that geometry in neural representation is not merely incidental, but is in fact the proper object for enabling principled control via intervention on internals. This recasts the core problem of steering from finding the right direction to finding the right geometry.

In this paper we show that visual diffusion models can serve as effective geometric solvers: they can directly reason about geometric problems by working in pixel space. We first demonstrate this on the Inscribed Square Problem, a long-standing problem in geometry that asks whether every Jordan curve contains four points forming a square. We then extend the approach to two other well-known hard geometric problems: the Steiner Tree Problem and the Simple Polygon Problem. Our method treats each problem instance as an image and trains a standard visual diffusion model that transforms Gaussian noise into an image representing a valid approximate solution that closely matches the exact one. The model learns to transform noisy geometric structures into correct configurations, effectively recasting geometric reasoning as image generation. Unlike prior work that necessitates specialized architectures and domain-specific adaptations when applying diffusion to parametric geometric representations, we employ a standard visual diffusion model that operates on the visual representation of the problem. This simplicity highlights a surprising bridge between generative modeling and geometric problem solving. Beyond the specific problems studied here, our results point toward a broader paradigm: operating in image space provides a general and practical framework for approximating notoriously hard problems, and opens the door to tackling a far wider class of challenging geometric tasks.

Recent advances in generative models highlight the power of geometry-aware modeling in manifold-constrained settings. Yet, for natural images, the field remains confined to Euclidean assumptions, failing to exploit the potential of intrinsic geometric structures within the data. In this work, we investigate the geometry of natural images and observe that semantic information is predominantly encoded in directional components, while norm components can be approximated by the global average. This property holds across both RGB and latent spaces, suggesting that natural images can be effectively modeled on a hypersphere. Building on this finding, we introduce Spherical Optimal Transport Flow Matching (SOT-CFM), which utilizes angular distance, and Spherical Flow Matching (SFM), which constrains dynamics directly on the manifold. Our experiments demonstrate that these geometry-aware methods achieve superior performance against Euclidean baselines. Ultimately, this work provides a novel perspective that bridges the gap between Riemannian manifold-based modeling and natural image generation.

Multimodal large language models (MLLMs) have made significant progress in integrating visual and linguistic understanding. Existing benchmarks typically focus on high-level semantic capabilities, such as scene understanding and visual reasoning, but often overlook a crucial, foundational ability: geometric perception. Geometric perception involves understanding geometric shapes, structures, and spatial relationships, which are essential for supporting higher-level semantic tasks. Despite its importance, this capability remains underexplored in current MLLM research. To address this gap, we introduce GePBench, a novel benchmark designed to assess the geometric perception abilities of MLLMs. Our extensive evaluations reveal that current state-of-the-art MLLMs exhibit significant deficiencies in geometric perception tasks. Furthermore, we show that models trained with GePBench data demonstrate substantial improvements on a wide range of benchmark tasks, highlighting the critical role of geometric perception in enabling advanced multimodal applications. Our code and datasets will be publicly available.

Mathematics olympiads are prestigious competitions, with problem proposing and solving highly honored. Building artificial intelligence that proposes and solves olympiads presents an unresolved challenge in automated theorem discovery and proving, especially in geometry for its combination of numerical and spatial elements. We introduce TongGeometry, a Euclidean geometry system supporting tree-search-based guided problem proposing and solving. The efficient geometry system establishes the most extensive repository of geometry theorems to date: within the same computational budget as the existing state-of-the-art, TongGeometry discovers 6.7 billion geometry theorems requiring auxiliary constructions, including 4.1 billion exhibiting geometric symmetry. Among them, 10 theorems were proposed to regional mathematical olympiads with 3 of TongGeometry's proposals selected in real competitions, earning spots in a national team qualifying exam or a top civil olympiad in China and the US. Guided by fine-tuned large language models, TongGeometry solved all International Mathematical Olympiad geometry in IMO-AG-30, outperforming gold medalists for the first time. It also surpasses the existing state-of-the-art across a broader spectrum of olympiad-level problems. The full capabilities of the system can be utilized on a consumer-grade machine, making the model more accessible and fostering widespread democratization of its use. By analogy, unlike existing systems that merely solve problems like students, TongGeometry acts like a geometry coach, discovering, presenting, and proving theorems.

This paper concerns the question of how AI systems encode semantic structure into the geometric structure of their representation spaces. The motivating observation of this paper is that the natural geometry of these representation spaces should reflect the way models use representations to produce behavior. We focus on the important special case of representations that define softmax distributions. In this case, we argue that the natural geometry is information geometry. Our focus is on the role of information geometry on semantic encoding and the linear representation hypothesis. As an illustrative application, we develop "dual steering", a method for robustly steering representations to exhibit a particular concept using linear probes. We prove that dual steering optimally modifies the target concept while minimizing changes to off-target concepts. Empirically, we find that dual steering enhances the controllability and stability of concept manipulation.

Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Euclidean spaces. However, many real-world problems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to suboptimal generative performance. To address this limitation, we introduce HyperSphereDiff to align hyperspherical structures with directional noise, preserving class geometry and effectively capturing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic geometry of hyperspherical data, resulting in more accurate and geometry-aware generative models. We evaluate our framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underlying hyperspherical manifold. Resources are available at: {https://github.com/IAB-IITJ/Harmonizing-Geometry-and-Uncertainty-Diffusion-with-Hyperspheres/}

Shape assembly aims to reassemble parts (or fragments) into a complete object, which is a common task in our daily life. Different from the semantic part assembly (e.g., assembling a chair's semantic parts like legs into a whole chair), geometric part assembly (e.g., assembling bowl fragments into a complete bowl) is an emerging task in computer vision and robotics. Instead of semantic information, this task focuses on geometric information of parts. As the both geometric and pose space of fractured parts are exceptionally large, shape pose disentanglement of part representations is beneficial to geometric shape assembly. In our paper, we propose to leverage SE(3) equivariance for such shape pose disentanglement. Moreover, while previous works in vision and robotics only consider SE(3) equivariance for the representations of single objects, we move a step forward and propose leveraging SE(3) equivariance for representations considering multi-part correlations, which further boosts the performance of the multi-part assembly. Experiments demonstrate the significance of SE(3) equivariance and our proposed method for geometric shape assembly. Project page: https://crtie.github.io/SE-3-part-assembly/

We propose a Geometry-aware Policy Imitation (GPI) approach that rethinks imitation learning by treating demonstrations as geometric curves rather than collections of state-action samples. From these curves, GPI derives distance fields that give rise to two complementary control primitives: a progression flow that advances along expert trajectories and an attraction flow that corrects deviations. Their combination defines a controllable, non-parametric vector field that directly guides robot behavior. This formulation decouples metric learning from policy synthesis, enabling modular adaptation across low-dimensional robot states and high-dimensional perceptual inputs. GPI naturally supports multimodality by preserving distinct demonstrations as separate models and allows efficient composition of new demonstrations through simple additions to the distance field. We evaluate GPI in simulation and on real robots across diverse tasks. Experiments show that GPI achieves higher success rates than diffusion-based policies while running 20 times faster, requiring less memory, and remaining robust to perturbations. These results establish GPI as an efficient, interpretable, and scalable alternative to generative approaches for robotic imitation learning. Project website: https://yimingli1998.github.io/projects/GPI/

The capacity to precisely edit genomes has outpaced our ability to predict the consequences. A cell can be genetically perfect and therapeutically useless: edited exactly as intended, yet unstable, drifting toward unintended fates, or selected for properties that compromise safety. This paradox reflects a deeper gap in how we evaluate biological intervention. Current frameworks excel at measuring what was done to a cell but remain blind to what the cell has become. We argue that this blindness stems from treating cells as collections of independent variables rather than as dynamical systems occupying positions on high-dimensional state manifolds. Drawing on Waddington's epigenetic landscape, we propose geometric stability as a missing axis of evaluation: the directional coherence of cellular responses to perturbation. This metric distinguishes interventions that guide cells coherently toward stable states from those that scatter them across the state manifold. Validation across diverse perturbation datasets reveals that geometric stability captures regulatory architecture invisible to conventional metrics, discriminating pleiotropic master regulators from lineage-specific factors without prior biological annotation. As precision medicine increasingly relies on cellular reprogramming, the question shifts from ``did the intervention occur?'' to ``is the resulting state stable?'' Geometric stability provides a framework for answering.

Videos inherently represent 2D projections of a dynamic 3D world. However, our analysis suggests that video diffusion models trained solely on raw video data often fail to capture meaningful geometric-aware structure in their learned representations. To bridge this gap between video diffusion models and the underlying 3D nature of the physical world, we propose Geometry Forcing, a simple yet effective method that encourages video diffusion models to internalize latent 3D representations. Our key insight is to guide the model's intermediate representations toward geometry-aware structure by aligning them with features from a pretrained geometric foundation model. To this end, we introduce two complementary alignment objectives: Angular Alignment, which enforces directional consistency via cosine similarity, and Scale Alignment, which preserves scale-related information by regressing unnormalized geometric features from normalized diffusion representation. We evaluate Geometry Forcing on both camera view-conditioned and action-conditioned video generation tasks. Experimental results demonstrate that our method substantially improves visual quality and 3D consistency over the baseline methods. Project page: https://GeometryForcing.github.io.

In this paper, we deal with the gluing of two surfaces, where the gluing locus is assumed to be a curve. We consider a moving frame along the gluing locus, and define developable surfaces with respect to the frame. Considering geometric properties of these developable surfaces, we study the geometry of gluing two surfaces.

Large language models (LLMs) generalize smoothly across continuous semantic spaces, yet strict logical reasoning demands the formation of discrete decision boundaries. Prevailing theories relying on linear isometric projections fail to resolve this fundamental tension. In this work, we argue that task context operates as a non-isometric dynamical operator that enforces a necessary "topological distortion." By applying Gram-Schmidt decomposition to residual-stream activations , we reveal a dual-modulation mechanism driving this process: a class-agnostic topological preservation that anchors global structure to prevent semantic collapse, and a specific algebraic divergence that directionally tears apart cross-class concepts to forge logical boundaries. We validate this geometric evolution across a gradient of tasks, from simple mapping to complex primality testing. Crucially, targeted specific vector ablation establishes a strict causal binding between this topology and model function: algebraically erasing the divergence component collapses parity classification accuracy from 100% to chance levels (38.57%). Furthermore, we uncover a three-phase layer-wise geometric dynamic and demonstrate that under social pressure prompts, models fail to generate sufficient divergence. This results in a "manifold entanglement" that geometrically explains sycophancy and hallucination. Ultimately, our findings revise the linear-isometric presumption, demonstrating that the emergence of discrete logic in LLMs is purchased at an irreducible cost of topological deformation.

Generative models have shown great promise in generating 3D geometric systems, which is a fundamental problem in many natural science domains such as molecule and protein design. However, existing approaches only operate on static structures, neglecting the fact that physical systems are always dynamic in nature. In this work, we propose geometric trajectory diffusion models (GeoTDM), the first diffusion model for modeling the temporal distribution of 3D geometric trajectories. Modeling such distribution is challenging as it requires capturing both the complex spatial interactions with physical symmetries and temporal correspondence encapsulated in the dynamics. We theoretically justify that diffusion models with equivariant temporal kernels can lead to density with desired symmetry, and develop a novel transition kernel leveraging SE(3)-equivariant spatial convolution and temporal attention. Furthermore, to induce an expressive trajectory distribution for conditional generation, we introduce a generalized learnable geometric prior into the forward diffusion process to enhance temporal conditioning. We conduct extensive experiments on both unconditional and conditional generation in various scenarios, including physical simulation, molecular dynamics, and pedestrian motion. Empirical results on a wide suite of metrics demonstrate that GeoTDM can generate realistic geometric trajectories with significantly higher quality.

Video diffusion models lack explicit geometric supervision during training, leading to inconsistency artifacts such as object deformation, spatial drift, and depth violations in generated videos. To address this limitation, we propose a geometry-based reward model that leverages pretrained geometric foundation models to evaluate multi-view consistency through cross-frame reprojection error. Unlike previous geometric metrics that measure inconsistency in pixel space, where pixel intensity may introduce additional noise, our approach conducts error computation in a pointwise fashion, yielding a more physically grounded and robust error metric. Furthermore, we introduce a geometry-aware sampling strategy that filters out low-texture and non-semantic regions, focusing evaluation on geometrically meaningful areas with reliable correspondences to improve robustness. We apply this reward model to align video diffusion models through two complementary pathways: post-training of a bidirectional model via SFT or Reinforcement Learning and inference-time optimization of a Causal Video Model (e.g., Streaming video generator) via test-time scaling with our reward as a path verifier. Experimental results validate the effectiveness of our design, demonstrating that our geometry-based reward provides superior robustness compared to other variants. By enabling efficient inference-time scaling, our method offers a practical solution for enhancing open-source video models without requiring extensive computational resources for retraining.

Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of geometric deep learning. Often problems in the physical sciences deal with relatively small sets of points in two- or three-dimensional space wherein translation, rotation, and permutation equivariance are important or even vital for models to be useful in practice. In this work, we present rotation- and permutation-equivariant architectures for deep learning on these small point clouds, composed of a set of products of terms from the geometric algebra and reductions over those products using an attention mechanism. The geometric algebra provides valuable mathematical structure by which to combine vector, scalar, and other types of geometric inputs in a systematic way to account for rotation invariance or covariance, while attention yields a powerful way to impose permutation equivariance. We demonstrate the usefulness of these architectures by training models to solve sample problems relevant to physics, chemistry, and biology.

Despite recent advances in multimodal reasoning, representing auxiliary geometric constructions remains a fundamental challenge for multimodal large language models (MLLMs). Such constructions are absent from the original diagram and must be introduced before theorems apply. Existing approaches predominantly rely on explicit construction paradigms, including text-based geometric specification, visual-token interleaving during reasoning, and tool-augmented geometric execution. However, these methods either fail to faithfully represent complex spatial relationships, incur representation mismatch between discrete symbols and continuous geometric structures, or rely on external capabilities that hinder end-to-end optimization. To address these limitations, we propose LatentGeo, a framework that learns continuous latent visual representations to internalize auxiliary geometric constructions without pixel-level rendering or external executors. We design a three-stage curriculum that progressively aligns and internalizes these latent representations through auxiliary visual supervision, followed by LaGDPO, a latent-aware reinforcement learning procedure that stabilizes latent representations during policy optimization while improving end-task correctness. To systematically evaluate construction-centric representation quality, we introduce GeoAux, a new benchmark targeting visually dependent geometry problems, and conduct experiments on GeoAux and MathVerse. Results show that LatentGeo achieves substantial gains on geometric reasoning tasks, particularly those requiring auxiliary constructions. Extensive analyses and ablation studies further validate the effectiveness of each component in our framework.

Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" is compact, structured, and low-dimensional. We formalize this principle through a deterministic functional-topological framework in which the set of valid realizations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite Hausdorff radius, and an induced continuous perceptual functional. This geometry provides explicit limits on knowledge, conditions for identifiability, and guarantees for generalization from sparse evidence -- properties fundamental to both natural and artificial intelligence. Across electromechanical, electrochemical, and physiological domains, we show that real-world processes consistently generate compact perceptual manifolds with the same geometric characteristics. Their boundaries can be discovered in a fully self-supervised manner as the empirical radius saturates with increasing sampling, even when the governing equations are unknown. These results demonstrate that deterministic functional topology offers a unified mathematical foundation for perception, representation, and world-model construction. It provides a geometric explanation for why biological learners and self-supervised AI systems can generalize from few observations, and establishes compact perceptual manifolds as a fundamental building block for future AI architectures. Finally, this work unifies biological perception and modern self-supervised models under a single geometric principle: both derive their generalization ability from the compactness and invariants of real-world perceptual manifolds.

The advent of Unified Multimodal Models (UMMs) signals a paradigm shift in artificial intelligence, moving from passive perception to active, cross-modal generation. Despite their unprecedented ability to synthesize information, a critical gap persists in evaluation: existing benchmarks primarily assess discriminative understanding or unconstrained image generation separately, failing to measure the integrated cognitive process of generative reasoning. To bridge this gap, we propose that geometric construction provides an ideal testbed as it inherently demands a fusion of language comprehension and precise visual generation. We introduce GGBench, a benchmark designed specifically to evaluate geometric generative reasoning. It provides a comprehensive framework for systematically diagnosing a model's ability to not only understand and reason but to actively construct a solution, thereby setting a more rigorous standard for the next generation of intelligent systems. Project website: https://opendatalab-raiser.github.io/GGBench/.

Large-scale video diffusion models achieve impressive visual quality, yet often fail to preserve geometric consistency. Prior approaches improve consistency either by augmenting the generator with additional modules or applying geometry-aware alignment. However, architectural modifications can compromise the generalization of internet-scale pretrained models, while existing alignment methods are limited to static scenes and rely on RGB-space rewards that require repeated VAE decoding, incurring substantial compute overhead and failing to generalize to highly dynamic real-world scenes. To preserve the pretrained capacity while improving geometric consistency, we propose VGGRPO (Visual Geometry GRPO), a latent geometry-guided framework for geometry-aware video post-training. VGGRPO introduces a Latent Geometry Model (LGM) that stitches video diffusion latents to geometry foundation models, enabling direct decoding of scene geometry from the latent space. By constructing LGM from a geometry model with 4D reconstruction capability, VGGRPO naturally extends to dynamic scenes, overcoming the static-scene limitations of prior methods. Building on this, we perform latent-space Group Relative Policy Optimization with two complementary rewards: a camera motion smoothness reward that penalizes jittery trajectories, and a geometry reprojection consistency reward that enforces cross-view geometric coherence. Experiments on both static and dynamic benchmarks show that VGGRPO improves camera stability, geometry consistency, and overall quality while eliminating costly VAE decoding, making latent-space geometry-guided reinforcement an efficient and flexible approach to world-consistent video generation.

Data heterogeneity in federated learning, characterized by a significant misalignment between local and global distributions, leads to divergent local optimization directions and hinders global model training. Existing studies mainly focus on optimizing local updates or global aggregation, but these indirect approaches demonstrate instability when handling highly heterogeneous data distributions, especially in scenarios where label skew and domain skew coexist. To address this, we propose a geometry-guided data generation method that centers on simulating the global embedding distribution locally. We first introduce the concept of the geometric shape of an embedding distribution and then address the challenge of obtaining global geometric shapes under privacy constraints. Subsequently, we propose GGEUR, which leverages global geometric shapes to guide the generation of new samples, enabling a closer approximation to the ideal global distribution. In single-domain scenarios, we augment samples based on global geometric shapes to enhance model generalization; in multi-domain scenarios, we further employ class prototypes to simulate the global distribution across domains. Extensive experimental results demonstrate that our method significantly enhances the performance of existing approaches in handling highly heterogeneous data, including scenarios with label skew, domain skew, and their coexistence. Code published at: https://github.com/WeiDai-David/2025CVPR_GGEUR

Large language models have seen widespread adoption in math problem-solving. However, in geometry problems that usually require visual aids for better understanding, even the most advanced multi-modal models currently still face challenges in effectively using image information. High-quality data is crucial for enhancing the geometric capabilities of multi-modal models, yet existing open-source datasets and related efforts are either too challenging for direct model learning or suffer from misalignment between text and images. To overcome this issue, we introduce a novel pipeline that leverages GPT-4 and GPT-4V to generate relatively basic geometry problems with aligned text and images, facilitating model learning. We have produced a dataset of 4.9K geometry problems and combined it with 19K open-source data to form our GeoGPT4V dataset. Experimental results demonstrate that the GeoGPT4V dataset significantly improves the geometry performance of various models on the MathVista and MathVision benchmarks. The code is available at https://github.com/Lanyu0303/GeoGPT4V_Project

Generative models have achieved success in producing semantically plausible 2D images, but it remains challenging in 3D generation due to the absence of spatial geometry constraints. Typically, existing methods utilize geometric features as conditions to enhance spatial awareness. However, these methods can only model relative relationships and are prone to scale inconsistency of absolute geometry. Thus, we argue that semantic information and absolute geometry empower 3D cognition, thereby enabling controllable 3D generation for the physical world. In this work, we propose Cog2Gen3D, a 3D cognition-guided diffusion framework for 3D generation. Our model is guided by three key designs: 1) Cognitive Feature Embeddings. We encode different modalities into semantic and geometric representations and further extract logical representations. 2) 3D Latent Cognition Graph. We structure different representations into dual-stream semantic-geometric graphs and fuse them via common-based cross-attention to obtain a 3D cognition graph. 3) Cognition-Guided Latent Diffusion. We leverage the fused 3D cognition graph as the condition to guide the latent diffusion process for 3D Gaussian generation. Under this unified framework, the 3D cognition graph ensures the physical plausibility and structural rationality of 3D generation. Moreover, we construct a validation subset based on the Marble World Labs. Extensive experiments demonstrate that our Cog2Gen3D significantly outperforms existing methods in both semantic fidelity and geometric plausibility.

We characterise sets of points of exceptional Lie incidence geometries, that is, the natural geometries arising from spherical buildings of exceptional types F_4, E_6, E_7, E_8 and G_2, that form a line using the opposition relation. With that, we obtain a classification of so-called ``geometric lines'' in many of these geometries. Furthermore, our results lead to a characterisation of geometric lines in finite exceptional Lie incidence geometries as minimal blocking sets, that is, point sets of the size of a line admitting no object opposite to all of their members, in most cases, and we classify all exceptions. As a further consequence, we obtain a characterisation of automorphisms of exceptional spherical buildings as certain opposition preserving maps.

Generative methods for 3D assets have recently achieved remarkable progress, yet providing intuitive and precise control over the object geometry remains a key challenge. Existing approaches predominantly rely on text or image prompts, which often fall short in geometric specificity: language can be ambiguous, and images are cumbersome to edit. In this work, we introduce SpaceControl, a training-free test-time method for explicit spatial control of 3D generation. Our approach accepts a wide range of geometric inputs, from coarse primitives to detailed meshes, and integrates seamlessly with modern pre-trained generative models without requiring any additional training. A controllable parameter lets users trade off between geometric fidelity and output realism. Extensive quantitative evaluation and user studies demonstrate that SpaceControl outperforms both training-based and optimization-based baselines in geometric faithfulness while preserving high visual quality. Finally, we present an interactive user interface that enables online editing of superquadrics for direct conversion into textured 3D assets, facilitating practical deployment in creative workflows. Find our project page at https://spacecontrol3d.github.io/

Vision Language Models (VLMs) exhibit a fundamental semantic-to-geometric gap in spatial reasoning: they excel at qualitative semantic inference but their reasoning operates within a lossy semantic space, misaligned with high-fidelity geometry. Current paradigms fail to bridge this gap. Training-based methods suffer from an ``oracle paradox,'' learning flawed spatial logic from imperfect oracles. Tool-integrated methods constrain the final computation but critically leave the VLM's planning process unconstrained, resulting in geometrically flawed plans. In this work, we propose Geometrically-Constrained Agent (GCA), a training-free agentic paradigm that resolves this gap by introducing a formal task constraint. Specifically, we strategically decouples the VLM's role into two stages. First, acting as a semantic analyst, the VLM translates the user's ambiguous query into the formal, verifiable task constraint, which defines the reference frame and objective. Second, acting as a task solver, the VLM generates and executes tool calls strictly within the deterministic bounds defined by the constraint. This geometrically-constrained reasoning strategy successfully resolve the semantic-to-geometric gap, yielding a robust and verifiable reasoning pathway for spatial reasoning. Comprehensive experiments demonstrate that GCA achieves SOTA performance on multiple spatial reasoning benchmarks, surpassing existing training-based and tool-integrated methods by ~27%. Please see our homepage at https://gca-spatial-reasoning.github.io.

Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly available tool to manipulate knot diagrams in a real-time, interactive way is yet to be developed. We introduce a method of operating on the geometry of the knot diagram itself without any underlying three-dimensional structure that can underpin such an application. This allows us to directly interact with vector graphics knot diagrams while at the same time computing knot invariants in ways proposed by previous work. An implementation of this method is provided.

The dominant paradigm for high-fidelity 3D generation relies on a VAE-Diffusion pipeline, where the VAE's reconstruction capability sets a firm upper bound on generation quality. A fundamental challenge limiting existing VAEs is the representation mismatch between ground-truth meshes and network predictions: GT meshes have arbitrary, variable topology, while VAEs typically predict fixed-structure implicit fields (\eg, SDF on regular grids). This inherent misalignment prevents establishing explicit mesh-level correspondences, forcing prior work to rely on indirect supervision signals such as SDF or rendering losses. Consequently, fine geometric details, particularly sharp features, are poorly preserved during reconstruction. To address this, we introduce TopoMesh, a sparse voxel-based VAE that unifies both GT and predicted meshes under a shared Dual Marching Cubes (DMC) topological framework. Specifically, we convert arbitrary input meshes into DMC-compliant representations via a remeshing algorithm that preserves sharp edges using an Linfty distance metric. Our decoder outputs meshes in the same DMC format, ensuring that both predicted and target meshes share identical topological structures. This establishes explicit correspondences at the vertex and face level, allowing us to derive explicit mesh-level supervision signals for topology, vertex positions, and face orientations with clear gradients. Our sparse VAE architecture employs this unified framework and is trained with Teacher Forcing and progressive resolution training for stable and efficient convergence. Extensive experiments demonstrate that TopoMesh significantly outperforms existing VAEs in reconstruction fidelity, achieving superior preservation of sharp features and geometric details.

This paper introduces a new interpretation of the Variational Autoencoder framework by taking a fully geometric point of view. We argue that vanilla VAE models unveil naturally a Riemannian structure in their latent space and that taking into consideration those geometrical aspects can lead to better interpolations and an improved generation procedure. This new proposed sampling method consists in sampling from the uniform distribution deriving intrinsically from the learned Riemannian latent space and we show that using this scheme can make a vanilla VAE competitive and even better than more advanced versions on several benchmark datasets. Since generative models are known to be sensitive to the number of training samples we also stress the method's robustness in the low data regime.

Graph foundation models represent a transformative paradigm for learning transferable representations across diverse graph domains. Recent methods leverage large language models to unify graph and text modalities into a shared representation space using contrastive learning. However, systematic evaluations reveal significant performance degradation at structural boundaries where distinct topological patterns converge, with accuracy losses exceeding 20 percentage points. This issue arises from a key limitation: current methods assume all graph structures can be encoded within a single Euclidean space. In reality, tree structures require hyperbolic geometry to preserve hierarchical branching, while cyclic patterns depend on spherical geometry for closure properties. At structural boundaries, nodes experience conflicting geometric constraints that uniform encoding spaces cannot resolve. This raises a crucial challenge: Can alignment frameworks be designed to respect the intrinsic geometric diversity of graph structures? We introduce GraphShaper, a geometry-aware framework that enhances graph encoding through multi-geometric specialization. Our approach employs expert networks tailored to different geometric spaces, dynamically computing fusion weights to adaptively integrate geometric properties based on local structural characteristics. This adaptive fusion preserves structural integrity before alignment with text embeddings. Extensive experiments demonstrate that GraphShaper achieves 9.47\% accuracy improvements on citation networks and 7.63\% on social networks in zero-shot settings.

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data, and on estimating local Hessian matrices of neural network loss landscapes.

Large Multimodal Models (LMMs) often struggle with geometric reasoning due to visual hallucinations and a lack of mathematically precise Chain-of-Thought (CoT) data. To address this, we propose the GeoSym Engine, an automated and scalable neuro-symbolic framework. By leveraging a type-conditional grammar and an analytic SymGT Solver, it derives exact symbolic ground truths and seamlessly integrates with a robust rendering pipeline to produce high-precision geometric diagrams. Using this engine, we construct GeoSym127K, a difficulty-stratified dataset featuring 51K high-resolution images, 127K questions with symbolic ground truths, and 55K answer-verified CoT QA pairs. We also introduce GeoSym-Bench, an expert-curated suite of 511 complex samples for rigorous evaluation. Through extensive supervised fine-tuning (SFT), we demonstrate that GeoSym drives concentrated improvements specifically on diagram-dependent and multi-step geometry tasks. Our Qwen3-VL-8B model gains an absolute +22.21% on the MathVerse Vision-Only subset and reaches 61.52% (+6.19% improvement) on WeMath, mitigating long-horizon logic fragmentation and outperforming advanced closed-source models like Doubao-1.8. Furthermore, applying Reinforcement Learning with Verifiable Rewards (RLVR) via GRPO reveals that initializing from structural SFT checkpoints substantially elevates the performance ceiling over zero-shot RL. Driven by deterministic exact-match signals, this showcases the robust scaling potential of our verifiable reasoning synthesis. Datasets and code are available at https://huggingface.co/datasets/Tomie0506/GeoSym127K and https://github.com/Tomie56/GeoSym127K.

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

Generating a street-level 3D scene from a single satellite image is a crucial yet challenging task. Current methods present a stark trade-off: geometry-colorization models achieve high geometric fidelity but are typically building-focused and lack semantic diversity. In contrast, proxy-based models use feed-forward image-to-3D frameworks to generate holistic scenes by jointly learning geometry and texture, a process that yields rich content but coarse and unstable geometry. We attribute these geometric failures to the extreme viewpoint gap and sparse, inconsistent supervision inherent in satellite-to-street data. We introduce Sat3DGen to address these fundamental challenges, which embodies a geometry-first methodology. This methodology enhances the feed-forward paradigm by integrating novel geometric constraints with a perspective-view training strategy, explicitly countering the primary sources of geometric error. This geometry-centric strategy yields a dramatic leap in both 3D accuracy and photorealism. For validation, we first constructed a new benchmark by pairing the VIGOR-OOD test set with high-resolution DSM data. On this benchmark, our method improves geometric RMSE from 6.76m to 5.20m. Crucially, this geometric leap also boosts photorealism, reducing the Fréchet Inception Distance (FID) from sim40 to 19 against the leading method, Sat2Density++, despite using no extra tailored image-quality modules. We demonstrate the versatility of our high-quality 3D assets through diverse downstream applications, including semantic-map-to-3D synthesis, multi-camera video generation, large-scale meshing, and unsupervised single-image Digital Surface Model (DSM) estimation. The code has been released on https://github.com/qianmingduowan/Sat3DGen.

Vision-Language Models (VLMs) frequently "hallucinate" - generate plausible yet factually incorrect statements - posing a critical barrier to their trustworthy deployment. In this work, we propose a new paradigm for diagnosing hallucinations, recasting them from static output errors into dynamic pathologies of a model's computational cognition. Our framework is grounded in a normative principle of computational rationality, allowing us to model a VLM's generation as a dynamic cognitive trajectory. We design a suite of information-theoretic probes that project this trajectory onto an interpretable, low-dimensional Cognitive State Space. Our central discovery is a governing principle we term the geometric-information duality: a cognitive trajectory's geometric abnormality within this space is fundamentally equivalent to its high information-theoretic surprisal. Hallucination detection is counts as a geometric anomaly detection problem. Evaluated across diverse settings - from rigorous binary QA (POPE) and comprehensive reasoning (MME) to unconstrained open-ended captioning (MS-COCO) - our framework achieves state-of-the-art performance. Crucially, it operates with high efficiency under weak supervision and remains highly robust even when calibration data is heavily contaminated. This approach enables a causal attribution of failures, mapping observable errors to distinct pathological states: perceptual instability (measured by Perceptual Entropy), logical-causal failure (measured by Inferential Conflict), and decisional ambiguity (measured by Decision Entropy). Ultimately, this opens a path toward building AI systems whose reasoning is transparent, auditable, and diagnosable by design.

Text-driven inversion of generative models is a core paradigm for manipulating 2D or 3D content, unlocking numerous applications such as text-based editing, style transfer, or inverse problems. However, it relies on the assumption that generative models remain sensitive to natural language prompts. We demonstrate that for state-of-the-art native text-to-3D generative models, this assumption often collapses. We identify a critical failure mode where generation trajectories are drawn into latent ``sink traps'': regions where the model becomes insensitive to prompt modifications. In these regimes, changes to the input text fail to alter internal representations in a way that alters the output geometry. Crucially, we observe that this is not a limitation of the model's geometric expressivity; the same generative models possess the ability to produce a vast diversity of shapes but, as we demonstrate, become insensitive to out-of-distribution text guidance. We investigate this behavior by analyzing the sampling trajectories of the generative model, and find that complex geometries can still be represented and produced by leveraging the model's unconditional generative prior. This leads to a more robust framework for text-based 3D shape editing that bypasses latent sinks by decoupling a model's geometric representation power from its linguistic sensitivity. Our approach addresses the limitations of current 3D pipelines and enables high-fidelity semantic manipulation of out-of-distribution 3D shapes. Project webpage: https://daidedou.sorpi.fr/publication/beyondprompts

Generalization, the ability to perform well beyond the training context, is a hallmark of biological and artificial intelligence, yet anticipating unseen failures remains a central challenge. Conventional approaches often take a ``bottom-up'' mechanistic route by reverse-engineering interpretable features or circuits to build explanatory models. While insightful, these methods often struggle to provide the high-level, predictive signals for anticipating failure in real-world deployment. Here, we propose using a ``top-down'' approach to studying generalization failures inspired by medical biomarkers: identifying system-level measurements that serve as robust indicators of a model's future performance. Rather than mapping out detailed internal mechanisms, we systematically design and test network markers to probe structure, function links, identify prognostic indicators, and validate predictions in real-world settings. In image classification, we find that task-relevant geometric properties of in-distribution (ID) object manifolds consistently forecast poor out-of-distribution (OOD) generalization. In particular, reductions in two geometric measures, effective manifold dimensionality and utility, predict weaker OOD performance across diverse architectures, optimizers, and datasets. We apply this finding to transfer learning with ImageNet-pretrained models. We consistently find that the same geometric patterns predict OOD transfer performance more reliably than ID accuracy. This work demonstrates that representational geometry can expose hidden vulnerabilities, offering more robust guidance for model selection and AI interpretability.

Memory systems can store vastly different amounts of information despite similar hardware constraints. Here, we show that superior spatial memory emerges from a discrete stiffening of hippocampal population geometry-a transition from disorganized to crystalline collective coding. Comparing food-caching chickadees to non-caching zebra finches, we found that the caching hippocampus maintains a topologically rigid, "crystalline" geometry with significantly higher geometric stability (Shesha 0.245 v 0.166) and nearly two-fold greater temporal coherence (Shesha 0.393 v 0.209), while the non-caching hippocampus resembles a disorganized "mist." This stability is actively constructed by synergistic circuit dynamics: excitatory neurons form the spatial scaffold while inhibitory populations contribute orthogonal decorrelation, a circuit motif in which excitatory and inhibitory populations occupy largely non-overlapping representational subspaces. A double dissociation with Valiant's Stable Memory Allocator, a model predicting that dedicated neuron ensembles underlie each memory, confirms this advantage reflects continuous topological organization rather than discrete neuron allocation: caching networks exhibit near-zero split-half allocation reliability despite their geometric superiority. Computational modeling across 10k configurations reveals topological rigidity as the mathematical prerequisite for scale: crystalline codes sustain high-fidelity readout beyond M=1k locations while mist codes fail below M=10, a >100-fold capacity advantage. This capacity requires a 169fold representational redundancy: a "geometric tax" stabilizing the manifold against biological noise. These results establish geometric stability as a candidate organizing principle of biological memory: evolution achieves high-capacity memory not by proliferating neurons, but by engineering the geometry of the neural code itself.

Sparse autoencoders (SAEs) are widely used to extract interpretable features from neural network representations, often under the implicit assumption that concepts correspond to independent linear directions. However, a growing body of evidence suggests that many concepts are instead organized along low-dimensional manifolds encoding continuous geometric relationships. This raises three basic questions: what does it mean for an SAE to capture a manifold, when do existing SAE architectures do so, and how? We develop a theoretical framework that answers these questions and show that SAEs can capture manifolds in two fundamentally different ways: globally, by allocating a compact group of atoms whose linear span contains the entire manifold, or locally, by distributing it across features that each selectively tile a restricted region of the underlying geometry. Empirically, we find that SAEs suboptimally recover continuous structures, mixing the global subspace and local tiling solutions in a fragmented regime we call dilution. This explains why manifold structure is rarely visible at the level of individual concepts and motivates post-hoc unsupervised discovery methods that search for coherent groups of atoms rather than isolated directions. More broadly, our results suggest that future representation learning methods should treat geometric objects, not just individual directions, as the basic units of interpretability.

Geometric spatial reasoning forms the foundation of many applications in artificial intelligence, yet the ability of large language models (LLMs) to operate over geometric spatial information expressed in procedural code remains underexplored. In this paper, we address this gap by formalizing the Program-to-Geometry task, which challenges models to translate programmatic drawing code into accurate and abstract geometric reasoning. To evaluate this capability, we present GeoGramBench, a benchmark of 500 carefully refined problems organized by a tailored three-level taxonomy that considers geometric complexity rather than traditional mathematical reasoning complexity. Our comprehensive evaluation of 17 frontier LLMs reveals consistent and pronounced deficiencies: even the most advanced models achieve less than 50% accuracy at the highest abstraction level. These results highlight the unique challenges posed by program-driven spatial reasoning and establish GeoGramBench as a valuable resource for advancing research in symbolic-to-spatial geometric reasoning. Project page: https://github.com/LiAuto-DSR/GeoGramBench.

Weight-perturbation evolution strategies (ES) can fine-tune billion-parameter language models with surprisingly small populations (e.g., N!approx!30), contradicting classical zeroth-order curse-of-dimensionality intuition. We also observe a second seemingly separate phenomenon: under fixed hyperparameters, the stochastic fine-tuning reward often rises, peaks, and then degrades in both ES and GRPO. We argue that both effects reflect a shared geometric property of fine-tuning landscapes: they are low-dimensional in curvature. A small set of high-curvature dimensions dominates improvement, producing (i) heterogeneous time scales that yield rise-then-decay under fixed stochasticity, as captured by a minimal quadratic stochastic-ascent model, and (ii) degenerate improving updates, where many random perturbations share similar components along these directions. Using ES as a geometric probe on fine-tuning reward landscapes of GSM8K, ARC-C, and WinoGrande across Qwen2.5-Instruct models (0.5B--7B), we show that reward-improving perturbations remain empirically accessible with small populations across scales. Together, these results reconcile ES scalability with non-monotonic training dynamics and suggest that high-dimensional fine-tuning may admit a broader class of viable optimization methods than worst-case theory implies.

Reliable deployment of language models requires two capabilities that appear distinct but share a common geometric foundation: predicting whether a model will accept targeted behavioral control, and detecting when its internal structure degrades. We show that geometric stability, the consistency of a representation's pairwise distance structure, addresses both. Supervised Shesha variants that measure task-aligned geometric stability predict linear steerability with near-perfect accuracy (ρ= 0.89-0.97) across 35-69 embedding models and three NLP tasks, capturing unique variance beyond class separability (partial ρ= 0.62-0.76). A critical dissociation emerges: unsupervised stability fails entirely for steering on real-world tasks (ρapprox 0.10), revealing that task alignment is essential for controllability prediction. However, unsupervised stability excels at drift detection, measuring nearly 2times greater geometric change than CKA during post-training alignment (up to 5.23times in Llama) while providing earlier warning in 73\% of models and maintaining a 6times lower false alarm rate than Procrustes. Together, supervised and unsupervised stability form complementary diagnostics for the LLM deployment lifecycle: one for pre-deployment controllability assessment, the other for post-deployment monitoring.

In this report, we introduce UltraShape 1.0, a scalable 3D diffusion framework for high-fidelity 3D geometry generation. The proposed approach adopts a two-stage generation pipeline: a coarse global structure is first synthesized and then refined to produce detailed, high-quality geometry. To support reliable 3D generation, we develop a comprehensive data processing pipeline that includes a novel watertight processing method and high-quality data filtering. This pipeline improves the geometric quality of publicly available 3D datasets by removing low-quality samples, filling holes, and thickening thin structures, while preserving fine-grained geometric details. To enable fine-grained geometry refinement, we decouple spatial localization from geometric detail synthesis in the diffusion process. We achieve this by performing voxel-based refinement at fixed spatial locations, where voxel queries derived from coarse geometry provide explicit positional anchors encoded via RoPE, allowing the diffusion model to focus on synthesizing local geometric details within a reduced, structured solution space. Our model is trained exclusively on publicly available 3D datasets, achieving strong geometric quality despite limited training resources. Extensive evaluations demonstrate that UltraShape 1.0 performs competitively with existing open-source methods in both data processing quality and geometry generation. All code and trained models will be released to support future research.

Large language models (LLMs) have shown remarkable proficiency in human-level reasoning and generation capabilities, which encourages extensive research on their application in mathematical problem solving. However, current work has been largely focused on text-based mathematical problems, with limited investigation in problems involving geometric information. Addressing this gap, we aim to enable LLMs to solve geometric problems by understanding image input. We first analyze the limitations of current Multimodal Large Language Models (MLLMs) in this area: they struggle to accurately comprehending basic geometric elements and their relationships. To overcome these challenges, we take advantage of the unique characteristics of geometric problems (such as unique geometric logical form, and geometric scalability) and the capacity of the textual LLMs to build an enriched multimodal geometry dataset based on existing data. The augmented dataset, Geo170K, contains more than 170K geometric image-caption and question-answer pairs. Utilizing our constructed Geo170K dataset, we develop G-LLaVA, which demonstrates exceptional performance in solving geometric problems, significantly outperforming GPT-4-V on the MathVista benchmark with only 7B parameters.

We revisit the geometric theory of defects. In the differential-geometric models of defects that have been adopted since the 1950s, dislocations have been associated with torsion, disclinations with the full curvature, and point defects with the first kind trace of non-metricity. The mainstream formulation exhibits several conceptual and technical shortcomings, most notably a hierarchy inconsistency, the non-exictence of a genuine metric formulation, and the potential emergence of Ostrogradsky-type instabilities. These issues have motivated us to develop a new framework, namely a generalized teleparallel geometric theory of defects. In our model, dislocations are identified with the trace of torsion, disclinations with the second kind trace of the non-metricity, and point defects with the first kind trace of the non-metricity. In addition, we retain the scalar part torsion as a free parameter for describing some possible unknown degrees of freedom in the theory of defects. The proposed geometric theory of defects is free from all of the aforementioned drawbacks and is therefore worthy of further investigation. To ensure the coherence and completeness of the discussion, we begin our analysis with elastic deformations, then summarize the existing metric-affine geometric theory of defects, and finally proceed to our original contribution, namely the new theory introduced here. We formulate the entire theory in Eulerian coordinates. Naturally, all results can be reformulated in Lagrangian coordinates as well. All analyses and formulae are expressed in the language of exterior algebra and are carried out in coordinate-independent orthonormal frames.

The discovery of extremal structures in mathematics requires navigating vast and nonconvex landscapes where analytical methods offer little guidance and brute-force search becomes intractable. We introduce FlowBoost, a closed-loop generative framework that learns to discover rare and extremal geometric structures by combining three components: (i) a geometry-aware conditional flow-matching model that learns to sample high-quality configurations, (ii) reward-guided policy optimization with action exploration that directly optimizes the generation process toward the objective while maintaining diversity, and (iii) stochastic local search for both training-data generation and final refinement. Unlike prior open-loop approaches, such as PatternBoost that retrains on filtered discrete samples, or AlphaEvolve which relies on frozen Large Language Models (LLMs) as evolutionary mutation operators, FlowBoost enforces geometric feasibility during sampling, and propagates reward signal directly into the generative model, closing the optimization loop and requiring much smaller training sets and shorter training times, and reducing the required outer-loop iterations by orders of magnitude, while eliminating dependence on LLMs. We demonstrate the framework on four geometric optimization problems: sphere packing in hypercubes, circle packing maximizing sum of radii, the Heilbronn triangle problem, and star discrepancy minimization. In several cases, FlowBoost discovers configurations that match or exceed the best known results. For circle packings, we improve the best known lower bounds, surpassing the LLM-based system AlphaEvolve while using substantially fewer computational resources.

Hamiltonian Monte Carlo has proven a remarkable empirical success, but only recently have we begun to develop a rigorous understanding of why it performs so well on difficult problems and how it is best applied in practice. Unfortunately, that understanding is confined within the mathematics of differential geometry which has limited its dissemination, especially to the applied communities for which it is particularly important. In this review I provide a comprehensive conceptual account of these theoretical foundations, focusing on developing a principled intuition behind the method and its optimal implementations rather of any exhaustive rigor. Whether a practitioner or a statistician, the dedicated reader will acquire a solid grasp of how Hamiltonian Monte Carlo works, when it succeeds, and, perhaps most importantly, when it fails.

Representational analysis explores how input data of a neural system are encoded in high dimensional spaces of its distributed neural activations, and how we can compare different systems, for instance, artificial neural networks and brains, on those grounds. While existing methods offer important insights, they typically do not account for local intrinsic geometrical properties within the high-dimensional representation spaces. To go beyond these limitations, we explore Ollivier-Ricci curvature and Ricci flow as tools to study the alignment of representations between humans and artificial neural systems on a geometric level. As a proof-of-principle study, we compared the representations of face stimuli between VGG-Face, a human-aligned version of VGG-Face, and corresponding human similarity judgments from a large online study. Using this discrete geometric framework, we were able to identify local structural similarities and differences by examining the distributions of node and edge curvature and higher-level properties by detecting and comparing community structure in the representational graphs.

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.

Modern monocular 3D reconstruction methods and vision-language models (VLMs) demonstrate impressive results on standard benchmarks, yet recent works cast doubt on their true understanding of geometric properties. We introduce GOQ, a comprehensive benchmark specifically designed to evaluate the geometric reasoning capabilities of vision and vision-language foundation models. GIQ comprises synthetic and real-world images and corresponding 3D meshes of diverse polyhedra covering varying levels of complexity and symmetry, from Platonic, Archimedean, Johnson, and Catalan solids to stellations and compound shapes. Through systematic experiments involving monocular 3D reconstruction, 3D symmetry detection, mental rotation tests, and zero-shot shape classification tasks, we reveal significant shortcomings in current models. State-of-the-art reconstruction algorithms trained on extensive 3D datasets struggle to reconstruct even basic geometric Platonic solids accurately. Next, although foundation models may be shown via linear and non-linear probing to capture specific 3D symmetry elements, they falter significantly in tasks requiring detailed geometric differentiation, such as mental rotation. Moreover, advanced vision-language assistants such as ChatGPT, Gemini and Claud exhibit remarkably low accuracy in interpreting basic shape properties such as face geometry, convexity, and compound structures of complex polyhedra. GIQ is publicly available at toomanymatts.github.io/giq-benchmark/, providing a structured platform to benchmark critical gaps in geometric intelligence and facilitate future progress in robust, geometry-aware representation learning.

In this paper, we present a deep learning-based framework for solving geometric construction problems through visual reasoning, which is useful for automated geometry theorem proving. Constructible problems in geometry often ask for the sequence of straightedge-and-compass constructions to construct a given goal given some initial setup. Our EuclidNet framework leverages the neural network architecture Mask R-CNN to extract the visual features from the initial setup and goal configuration with extra points of intersection, and then generate possible construction steps as intermediary data models that are used as feedback in the training process for further refinement of the construction step sequence. This process is repeated recursively until either a solution is found, in which case we backtrack the path for a step-by-step construction guide, or the problem is identified as unsolvable. Our EuclidNet framework is validated on complex Japanese Sangaku geometry problems, demonstrating its capacity to leverage backtracking for deep visual reasoning of challenging problems.

Faithful visualizations of data residing on manifolds must take the underlying geometry into account when producing a flat planar view of the data. In this paper, we extend the classic stochastic neighbor embedding (SNE) algorithm to data on general Riemannian manifolds. We replace standard Gaussian assumptions with Riemannian diffusion counterparts and propose an efficient approximation that only requires access to calculations of Riemannian distances and volumes. We demonstrate that the approach also allows for mapping data from one manifold to another, e.g. from a high-dimensional sphere to a low-dimensional one.

Problems involving geometric data arise in a variety of fields, including computer vision, robotics, chemistry, and physics. Such data can take numerous forms, such as points, direction vectors, planes, or transformations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric algebra, which offers an efficient 16-dimensional vector space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a transformer, GATr is scalable, expressive, and versatile. In experiments with n-body modeling and robotic planning, GATr shows strong improvements over non-geometric baselines.

Learning a shared policy that guides the locomotion of different agents is of core interest in Reinforcement Learning (RL), which leads to the study of morphology-agnostic RL. However, existing benchmarks are highly restrictive in the choice of starting point and target point, constraining the movement of the agents within 2D space. In this work, we propose a novel setup for morphology-agnostic RL, dubbed Subequivariant Graph RL in 3D environments (3D-SGRL). Specifically, we first introduce a new set of more practical yet challenging benchmarks in 3D space that allows the agent to have full Degree-of-Freedoms to explore in arbitrary directions starting from arbitrary configurations. Moreover, to optimize the policy over the enlarged state-action space, we propose to inject geometric symmetry, i.e., subequivariance, into the modeling of the policy and Q-function such that the policy can generalize to all directions, improving exploration efficiency. This goal is achieved by a novel SubEquivariant Transformer (SET) that permits expressive message exchange. Finally, we evaluate the proposed method on the proposed benchmarks, where our method consistently and significantly outperforms existing approaches on single-task, multi-task, and zero-shot generalization scenarios. Extensive ablations are also conducted to verify our design. Code and videos are available on our project page: https://alpc91.github.io/SGRL/.

Computer-aided design (CAD) is the digital construction of 2D and 3D objects, and is central to a wide range of engineering and manufacturing applications like automobile and aviation. Despite its importance, CAD modeling remains largely a time-intensive, manual task. Recent works have attempted to automate this process with small transformer-based models and handcrafted CAD sequence representations. However, there has been little effort to leverage the potential of large language models (LLMs) for sequential CAD design. In this work, we introduce a new large-scale dataset of more than 170k CAD models annotated with high-quality, human-like descriptions generated with our pipeline based on GPT-4.1. Using this dataset, we fine-tune powerful code-LLMs to generate CAD sequences represented in a JSON-based format from natural language descriptions, demonstrating the viability and effectiveness of this approach for text-conditioned CAD generation. Because simple metrics often fail to reflect the quality of generated objects, we introduce geometric and topological metrics based on sphericity, mean curvature, and Euler characteristic to provide richer structural insights. Our experiments and ablation studies on both synthetic and human-annotated data demonstrate that CADmium is able to automate CAD design, drastically speeding up the design of new objects. The dataset, code, and fine-tuned models are available online.

Multimodal large language models have various practical applications that demand strong reasoning abilities. Despite recent advancements, these models still struggle to solve complex geometric problems. A key challenge stems from the lack of high-quality image-text pair datasets for understanding geometric images. Furthermore, most template-based data synthesis pipelines typically fail to generalize to questions beyond their predefined templates. In this paper, we bridge this gap by introducing a complementary process of Reinforcement Learning with Verifiable Rewards (RLVR) into the data generation pipeline. By adopting RLVR to refine captions for geometric images synthesized from 50 basic geometric relations and using reward signals derived from mathematical problem-solving tasks, our pipeline successfully captures the key features of geometry problem-solving. This enables better task generalization and yields non-trivial improvements. Furthermore, even in out-of-distribution scenarios, the generated dataset enhances the general reasoning capabilities of multimodal large language models, yielding accuracy improvements of 2.8%-4.8% in statistics, arithmetic, algebraic, and numerical tasks with non-geometric input images of MathVista and MathVerse, along with 2.4%-3.9% improvements in Art, Design, Tech, and Engineering tasks in MMMU.

The creation of photorealistic virtual worlds requires the accurate modeling of 3D surface geometry for a wide range of objects. For this, meshes are appealing since they 1) enable fast physics-based rendering with realistic material and lighting, 2) support physical simulation, and 3) are memory-efficient for modern graphics pipelines. Recent work on reconstructing and statistically modeling 3D shape, however, has critiqued meshes as being topologically inflexible. To capture a wide range of object shapes, any 3D representation must be able to model solid, watertight, shapes as well as thin, open, surfaces. Recent work has focused on the former, and methods for reconstructing open surfaces do not support fast reconstruction with material and lighting or unconditional generative modelling. Inspired by the observation that open surfaces can be seen as islands floating on watertight surfaces, we parameterize open surfaces by defining a manifold signed distance field on watertight templates. With this parameterization, we further develop a grid-based and differentiable representation that parameterizes both watertight and non-watertight meshes of arbitrary topology. Our new representation, called Ghost-on-the-Shell (G-Shell), enables two important applications: differentiable rasterization-based reconstruction from multiview images and generative modelling of non-watertight meshes. We empirically demonstrate that G-Shell achieves state-of-the-art performance on non-watertight mesh reconstruction and generation tasks, while also performing effectively for watertight meshes.

Existing RGB-based imitation learning approaches typically employ traditional vision encoders such as ResNet or ViT, which lack explicit 3D reasoning capabilities. Recent geometry-grounded vision models, such as VGGT~wang2025vggt, provide robust spatial understanding and are promising candidates to address this limitation. This work investigates the integration of geometry-aware visual representations into robotic manipulation. Our results suggest that incorporating the geometry-aware vision encoder into imitation learning frameworks, including ACT and DP, yields up to 6.5% improvement over standard vision encoders in success rate across single- and bi-manual manipulation tasks in both simulation and real-world settings. Despite these benefits, most geometry-grounded models require high computational cost, limiting their deployment in practical robotic systems. To address this challenge, we propose eVGGT, an efficient geometry-aware encoder distilled from VGGT. eVGGT is nearly 9 times faster and 5 times smaller than VGGT, while preserving strong 3D reasoning capabilities. Code and pretrained models will be released to facilitate further research in geometry-aware robotics.

Recent advances in large language models (LLMs) have demonstrated remarkable capabilities across diverse domains, particularly in mathematical reasoning, amid which geometry problem solving remains a challenging area where auxiliary construction plays a enssential role. Existing approaches either achieve suboptimal performance or rely on massive LLMs (e.g., GPT-4o), incurring massive computational costs. We posit that reinforcement learning with verifiable reward (e.g., GRPO) offers a promising direction for training smaller models that effectively combine auxiliary construction with robust geometric reasoning. However, directly applying GRPO to geometric reasoning presents fundamental limitations due to its dependence on unconditional rewards, which leads to indiscriminate and counterproductive auxiliary constructions. To address these challenges, we propose Group Contrastive Policy Optimization (GCPO), a novel reinforcement learning framework featuring two key innovations: (1) Group Contrastive Masking, which adaptively provides positive or negative reward signals for auxiliary construction based on contextual utility, and a (2) length reward that promotes longer reasoning chains. Building on GCPO, we develop GeometryZero, a family of affordable-size geometric reasoning models that judiciously determine when to employ auxiliary construction. Our extensive empirical evaluation across popular geometric benchmarks (Geometry3K, MathVista) demonstrates that GeometryZero models consistently outperform baselines (e.g. GRPO), achieving an average improvement of 4.29% across all benchmarks.

Despite their success, modern convolutional neural networks (CNNs) exhibit fundamental limitations, including data inefficiency, poor out-of-distribution generalization, and vulnerability to adversarial perturbations. These shortcomings can be traced to a lack of inductive biases that reflect the inherent geometric structure of the visual world. The primate visual system, in contrast, demonstrates superior efficiency and robustness, suggesting that its architectural and computational principles,which evolved to internalize these structures,may offer a blueprint for more capable artificial vision. This paper introduces Visual Cortex Network (VCNet), a novel neural network architecture whose design is informed by the macro-scale organization of the primate visual cortex. VCNet is framed as a geometric framework that emulates key biological mechanisms, including hierarchical processing across distinct cortical areas, dual-stream information segregation for learning disentangled representations, and top-down predictive feedback for representation refinement. We interpret these mechanisms through the lens of geometry and dynamical systems, positing that they guide the learning of structured, low-dimensional neural manifolds. We evaluate VCNet on two specialized benchmarks: the Spots-10 animal pattern dataset, which probes sensitivity to natural textures, and a light field image classification task, which requires processing higher-dimensional visual data. Our results show that VCNet achieves state-of-the-art accuracy of 92.1\% on Spots-10 and 74.4\% on the light field dataset, surpassing contemporary models of comparable size. This work demonstrates that integrating high-level neuroscientific principles, viewed through a geometric lens, can lead to more efficient and robust models, providing a promising direction for addressing long-standing challenges in machine learning.

We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of x_T as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime z=(x_t,t) that indexes the family of denoising distributions p(x_0 | x_t) across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry.

Developing robust representations of chemical structures that enable models to learn topological inductive biases is challenging. In this manuscript, we present a representation of atomistic systems. We begin by proving that our representation satisfies all structural, geometric, efficiency, and generalizability constraints. Afterward, we provide a general algorithm to encode any atomistic system. Finally, we report performance comparable to state-of-the-art methods on numerous tasks. We open-source all code and datasets. The code and data are available at https://github.com/rahulkhorana/PolyatomicComplexes.

This paper presents the computational challenge on differential geometry and topology that happened within the ICLR 2021 workshop "Geometric and Topological Representation Learning". The competition asked participants to provide creative contributions to the fields of computational geometry and topology through the open-source repositories Geomstats and Giotto-TDA. The challenge attracted 16 teams in its two month duration. This paper describes the design of the challenge and summarizes its main findings.

We introduce TetSphere Splatting, a Lagrangian geometry representation designed for high-quality 3D shape modeling. TetSphere splatting leverages an underused yet powerful geometric primitive -- volumetric tetrahedral meshes. It represents 3D shapes by deforming a collection of tetrahedral spheres, with geometric regularizations and constraints that effectively resolve common mesh issues such as irregular triangles, non-manifoldness, and floating artifacts. Experimental results on multi-view and single-view reconstruction highlight TetSphere splatting's superior mesh quality while maintaining competitive reconstruction accuracy compared to state-of-the-art methods. Additionally, TetSphere splatting demonstrates versatility by seamlessly integrating into generative modeling tasks, such as image-to-3D and text-to-3D generation.

We tackle the problem of generating a complete vector map representation from aerial imagery in a single run: producing polygons for all land-cover classes with shared boundaries and without gaps or overlaps. Existing polygonization methods are typically class-specific; extending them to multiple classes via per-class runs commonly leads to topological inconsistencies, such as duplicated edges, gaps, and overlaps. We formalize this new task as All-Class Polygonal Vectorization (ACPV) and release the first public benchmark, Deventer-512, with standardized metrics jointly evaluating semantic fidelity, geometric accuracy, vertex efficiency, per-class topological fidelity and global topological consistency. To realize ACPV, we propose ACPV-Net, a unified framework introducing a novel Semantically Supervised Conditioning (SSC) mechanism coupling semantic perception with geometric primitive generation, along with a topological reconstruction that enforces shared-edge consistency by design. While enforcing such strict topological constraints, ACPV-Net surpasses all class-specific baselines in polygon quality across classes on Deventer-512. It also applies to single-class polygonal vectorization without any architectural modification, achieving the best-reported results on WHU-Building. Data, code, and models will be released at: https://github.com/HeinzJiao/ACPV-Net.

General object composition (GOC) aims to seamlessly integrate a target object into a background scene with desired geometric properties, while simultaneously preserving its fine-grained appearance details. Recent approaches derive semantic embeddings and integrate them into advanced diffusion models to enable geometry-editable generation. However, these highly compact embeddings encode only high-level semantic cues and inevitably discard fine-grained appearance details. We introduce a Disentangled Geometry-editable and Appearance-preserving Diffusion (DGAD) model that first leverages semantic embeddings to implicitly capture the desired geometric transformations and then employs a cross-attention retrieval mechanism to align fine-grained appearance features with the geometry-edited representation, facilitating both precise geometry editing and faithful appearance preservation in object composition. Specifically, DGAD builds on CLIP/DINO-derived and reference networks to extract semantic embeddings and appearance-preserving representations, which are then seamlessly integrated into the encoding and decoding pipelines in a disentangled manner. We first integrate the semantic embeddings into pre-trained diffusion models that exhibit strong spatial reasoning capabilities to implicitly capture object geometry, thereby facilitating flexible object manipulation and ensuring effective editability. Then, we design a dense cross-attention mechanism that leverages the implicitly learned object geometry to retrieve and spatially align appearance features with their corresponding regions, ensuring faithful appearance consistency. Extensive experiments on public benchmarks demonstrate the effectiveness of the proposed DGAD framework.

State-of-the-art video generation models produce remarkable photorealism, but they lack the precise control required to align generated content with specific scene requirements. Furthermore, without an underlying explicit geometry, these models cannot guarantee 3D consistency. Conversely, 3D engines offer granular control over every scene element and provide native 3D consistency by design, yet their output often remains trapped in the "uncanny valley". Bridging this sim-to-real gap requires both structural precision, where the output must exactly preserve the geometry and dynamics of the input, and global semantic transformation, where materials, lighting, and textures must be holistically transformed to achieve photorealism. We present RealMaster, a method that leverages video diffusion models to lift rendered video into photorealistic video while maintaining full alignment with the output of the 3D engine. To train this model, we generate a paired dataset via an anchor-based propagation strategy, where the first and last frames are enhanced for realism and propagated across the intermediate frames using geometric conditioning cues. We then train an IC-LoRA on these paired videos to distill the high-quality outputs of the pipeline into a model that generalizes beyond the pipeline's constraints, handling objects and characters that appear mid-sequence and enabling inference without requiring anchor frames. Evaluated on complex GTA-V sequences, RealMaster significantly outperforms existing video editing baselines, improving photorealism while preserving the geometry, dynamics, and identity specified by the original 3D control.

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.

We introduce a diffusion-based framework that performs aligned novel view image and geometry generation via a warping-and-inpainting methodology. Unlike prior methods that require dense posed images or pose-embedded generative models limited to in-domain views, our method leverages off-the-shelf geometry predictors to predict partial geometries viewed from reference images, and formulates novel-view synthesis as an inpainting task for both image and geometry. To ensure accurate alignment between generated images and geometry, we propose cross-modal attention distillation, where attention maps from the image diffusion branch are injected into a parallel geometry diffusion branch during both training and inference. This multi-task approach achieves synergistic effects, facilitating geometrically robust image synthesis as well as well-defined geometry prediction. We further introduce proximity-based mesh conditioning to integrate depth and normal cues, interpolating between point cloud and filtering erroneously predicted geometry from influencing the generation process. Empirically, our method achieves high-fidelity extrapolative view synthesis on both image and geometry across a range of unseen scenes, delivers competitive reconstruction quality under interpolation settings, and produces geometrically aligned colored point clouds for comprehensive 3D completion. Project page is available at https://cvlab-kaist.github.io/MoAI.

Recently, 3D Gaussian splatting (3D-GS) has achieved great success in reconstructing and rendering real-world scenes. To transfer the high rendering quality to generation tasks, a series of research works attempt to generate 3D-Gaussian assets from text. However, the generated assets have not achieved the same quality as those in reconstruction tasks. We observe that Gaussians tend to grow without control as the generation process may cause indeterminacy. Aiming at highly enhancing the generation quality, we propose a novel framework named GaussianDreamerPro. The main idea is to bind Gaussians to reasonable geometry, which evolves over the whole generation process. Along different stages of our framework, both the geometry and appearance can be enriched progressively. The final output asset is constructed with 3D Gaussians bound to mesh, which shows significantly enhanced details and quality compared with previous methods. Notably, the generated asset can also be seamlessly integrated into downstream manipulation pipelines, e.g. animation, composition, and simulation etc., greatly promoting its potential in wide applications. Demos are available at https://taoranyi.com/gaussiandreamerpro/.

Geometric diagrams are critical in conveying mathematical and scientific concepts, yet traditional diagram generation methods are often manual and resource-intensive. While text-to-image generation has made strides in photorealistic imagery, creating accurate geometric diagrams remains a challenge due to the need for precise spatial relationships and the scarcity of geometry-specific datasets. This paper presents MagicGeo, a training-free framework for generating geometric diagrams from textual descriptions. MagicGeo formulates the diagram generation process as a coordinate optimization problem, ensuring geometric correctness through a formal language solver, and then employs coordinate-aware generation. The framework leverages the strong language translation capability of large language models, while formal mathematical solving ensures geometric correctness. We further introduce MagicGeoBench, a benchmark dataset of 220 geometric diagram descriptions, and demonstrate that MagicGeo outperforms current methods in both qualitative and quantitative evaluations. This work provides a scalable, accurate solution for automated diagram generation, with significant implications for educational and academic applications.

We develop the GKM theory for the torus-equivariant cohomology of the affine flag variety using the combinatorics of alcove walks. Dual to the usual GKM setup, which depicts the orbits of the small torus action on a graph, alcove walks take place in tessellations of Euclidean space. Walks in affine rank two occur on triangulations of the plane, providing a more direct connection to splines used for approximating surfaces. Alcove walks in GKM theory also need not be minimal length, and can instead be randomly generated, giving rise to more flexible implementation. This work reinterprets and recovers classical results in GKM theory on the affine flag variety, generalizing them to both non-minimal and folded alcove walks, all motivated by applications to splines.

Reasoning-trained language models often spend more tokens on harder problems, but longer chains of thought do not show whether a model is merely computing for more steps or following a different internal trajectory. We study this distinction through hidden-state trajectories during chain-of-thought generation across competitive programming, mathematics, and Boolean satisfiability. Raw trajectory geometry is strongly shaped by generation length: longer generations mechanically alter path statistics, so difficulty-dependent comparisons are misleading without adjustment. After residualizing trajectory statistics on length, difficulty remains systematically coupled to corrected trajectory geometry across all domains studied. The clearest reasoning-specific separation appears in the code domain, where harder problems show more direct corrected trajectories and less heterogeneous local curvature in reasoning-trained models than in matched instruction-tuned baselines. Corrected difficulty-geometry coupling is weaker, but still present, in mathematics and Boolean satisfiability. Prompt-stage linear probes do not mirror the code-domain separation, and behavioral annotations show that stronger corrected coupling co-occurs with strategy shifts and uncertainty monitoring. Together, these findings establish length correction as a prerequisite for generation-time trajectory analysis and show that reasoning training can be associated with distinct corrected trajectory geometry, with the strength of the effect depending on the domain.

Geometry problems are a crucial testbed for AI reasoning capabilities. Most existing geometry solving systems cannot express problems within a unified framework, thus are difficult to integrate with other mathematical fields. Besides, since most geometric proofs rely on intuitive diagrams, verifying geometry problems is particularly challenging. To address these gaps, we introduce LeanGeo, a unified formal system for formalizing and solving competition-level geometry problems within the Lean 4 theorem prover. LeanGeo features a comprehensive library of high-level geometric theorems with Lean's foundational logic, enabling rigorous proof verification and seamless integration with Mathlib. We also present LeanGeo-Bench, a formal geometry benchmark in LeanGeo, comprising problems from the International Mathematical Olympiad (IMO) and other advanced sources. Our evaluation demonstrates the capabilities and limitations of state-of-the-art Large Language Models on this benchmark, highlighting the need for further advancements in automated geometric reasoning. We open source the theorem library and the benchmark of LeanGeo at https://github.com/project-numina/LeanGeo/tree/master.

Recent advances in flow-based offline reinforcement learning (RL) have achieved strong performance by parameterizing policies via flow matching. However, they still face critical trade-offs among expressiveness, optimality, and efficiency. In particular, existing flow policies interpret the L_2 regularization as an upper bound of the 2-Wasserstein distance (W_2), which can be problematic in offline settings. This issue stems from a fundamental geometric mismatch: the behavioral policy manifold is inherently anisotropic, whereas the L_2 (or upper bound of W_2) regularization is isotropic and density-insensitive, leading to systematically misaligned optimization directions. To address this, we revisit offline RL from a geometric perspective and show that policy refinement can be formulated as a local transport map: an initial flow policy augmented by a residual displacement. By analyzing the induced density transformation, we derive a local quadratic approximation of the KL-constrained objective governed by the Fisher information matrix, enabling a tractable anisotropic optimization formulation. By leveraging the score function embedded in the flow velocity, we obtain a corresponding quadratic constraint for efficient optimization. Our results reveal that the optimality gap in prior methods arises from their isotropic approximation. In contrast, our framework achieves a controllable approximation error within a provable neighborhood of the optimal solution. Extensive experiments demonstrate state-of-the-art performance across diverse offline RL benchmarks. See project page: https://github.com/ARC0127/Fisher-Decorator.

The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are invariant or equivariant to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at https://github.com/chaitjo/geometric-gnn-dojo

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

Multimodal Large Language Models (MLLMs) have achieved remarkable progress but continue to struggle with geometric reasoning, primarily due to the perception bottleneck regarding fine-grained visual elements. While formal languages have aided plane geometry understanding, solid geometry which requires spatial understanding remains largely unexplored. In this paper, we address this challenge by designing a unified formal language that integrates plane and solid geometry, comprehensively covering geometric structures and semantic relations. We construct GDP-29K, a large-scale dataset comprising 20k plane and 9k solid geometry samples collected from diverse real-world sources, each paired with its ground-truth formal description. To ensure syntactic correctness and geometric consistency, we propose a training paradigm that combines Supervised Fine-Tuning with Reinforcement Learning via Verifiable Rewards. Experiments show that our approach achieves state-of-the-art parsing performance. Furthermore, we demonstrate that our parsed formal descriptions serve as a critical cognitive scaffold, significantly boosting MLLMs' capabilities for downstream geometry reasoning tasks. Our data and code are available at Geoparsing.

We present DreamPolisher, a novel Gaussian Splatting based method with geometric guidance, tailored to learn cross-view consistency and intricate detail from textual descriptions. While recent progress on text-to-3D generation methods have been promising, prevailing methods often fail to ensure view-consistency and textural richness. This problem becomes particularly noticeable for methods that work with text input alone. To address this, we propose a two-stage Gaussian Splatting based approach that enforces geometric consistency among views. Initially, a coarse 3D generation undergoes refinement via geometric optimization. Subsequently, we use a ControlNet driven refiner coupled with the geometric consistency term to improve both texture fidelity and overall consistency of the generated 3D asset. Empirical evaluations across diverse textual prompts spanning various object categories demonstrate the efficacy of DreamPolisher in generating consistent and realistic 3D objects, aligning closely with the semantics of the textual instructions.

Multimodal large language models (MLLMs) have made rapid progress in recent years, yet continue to struggle with low-level visual perception (LLVP) -- particularly the ability to accurately describe the geometric details of an image. This capability is crucial for applications in areas such as robotics, medical image analysis, and manufacturing. In this paper, we first introduce Geoperception, a benchmark designed to evaluate an MLLM's ability to accurately transcribe 2D geometric information from an image. Using this benchmark, we demonstrate the limitations of leading MLLMs, and then conduct a comprehensive empirical study to explore strategies for improving their performance on geometric tasks. Our findings highlight the benefits of certain model architectures, training techniques, and data strategies, including the use of high-fidelity synthetic data and multi-stage training with a data curriculum. Notably, we find that a data curriculum enables models to learn challenging geometry understanding tasks which they fail to learn from scratch. Leveraging these insights, we develop Euclid, a family of models specifically optimized for strong low-level geometric perception. Although purely trained on synthetic multimodal data, Euclid shows strong generalization ability to novel geometry shapes. For instance, Euclid outperforms the best closed-source model, Gemini-1.5-Pro, by up to 58.56% on certain Geoperception benchmark tasks and 10.65% on average across all tasks.

While recent 3D generative models can produce high-quality texture images, they often fail to capture human preferences or meet task-specific requirements. Moreover, a core challenge in the 3D texture generation domain is that most existing approaches rely on repeated calls to 2D text-to-image generative models, which lack an inherent understanding of the 3D structure of the input 3D mesh object. To alleviate these issues, we propose an end-to-end differentiable, reinforcement-learning-free framework that embeds human feedback, expressed as differentiable reward functions, directly into the 3D texture synthesis pipeline. By back-propagating preference signals through both geometric and appearance modules of the proposed framework, our method generates textures that respect the 3D geometry structure and align with desired criteria. To demonstrate its versatility, we introduce three novel geometry-aware reward functions, which offer a more controllable and interpretable pathway for creating high-quality 3D content from natural language. By conducting qualitative, quantitative, and user-preference evaluations against state-of-the-art methods, we demonstrate that our proposed strategy consistently outperforms existing approaches. We will make our implementation code publicly available upon acceptance of the paper.

Geometry problem solving (GPS) represents a critical frontier in artificial intelligence, with profound applications in education, computer-aided design, and computational graphics. Despite its significance, automating GPS remains challenging due to the dual demands of spatial understanding and rigorous logical reasoning. Recent advances in large models have enabled notable breakthroughs, particularly for SAT-level problems, yet the field remains fragmented across methodologies, benchmarks, and evaluation frameworks. This survey systematically synthesizes GPS advancements through three core dimensions: (1) benchmark construction, (2) textual and diagrammatic parsing, and (3) reasoning paradigms. We further propose a unified analytical paradigm, assess current limitations, and identify emerging opportunities to guide future research toward human-level geometric reasoning, including automated benchmark generation and interpretable neuro-symbolic integration.

Neural networks often pack many unrelated concepts into a single neuron - a puzzling phenomenon known as 'polysemanticity' which makes interpretability much more challenging. This paper provides a toy model where polysemanticity can be fully understood, arising as a result of models storing additional sparse features in "superposition." We demonstrate the existence of a phase change, a surprising connection to the geometry of uniform polytopes, and evidence of a link to adversarial examples. We also discuss potential implications for mechanistic interpretability.

Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.

Plane Geometry Diagram Synthesis has been a crucial task in computer graphics, with applications ranging from educational tools to AI-driven mathematical reasoning. Traditionally, we rely on manual tools (e.g., Matplotlib and GeoGebra) to generate precise diagrams, but this usually requires huge, complicated calculations. Recently, researchers start to work on model-based methods (e.g., Stable Diffusion and GPT5) to automatically generate diagrams, saving operational cost but usually suffering from limited realism and insufficient accuracy. In this paper, we propose a novel framework GeoSDF, to automatically generate diagrams efficiently and accurately with Signed Distance Field (SDF). Specifically, we first represent geometric elements (e.g., points, segments, and circles) in the SDF, then construct a series of constraint functions to represent geometric relationships. Next, we optimize those constructed constraint functions to get an optimized field of both elements and constraints. Finally, by rendering the optimized field, we can obtain the synthesized diagram. In our GeoSDF, we define a symbolic language to represent geometric elements and constraints, and our synthesized geometry diagrams can be self-verified in the SDF, ensuring both mathematical accuracy and visual plausibility. In experiments, through both qualitative and quantitative analysis, GeoSDF synthesized both normal high-school level and IMO-level geometry diagrams. We achieve 88.67\% synthesis accuracy by human evaluation in the IMO problem set. Furthermore, we obtain a very high accuracy of solving geometry problems (over 95\% while the current SOTA accuracy is around 75%) by leveraging our self-verification property. All of these demonstrate the advantage of GeoSDF, paving the way for more sophisticated, accurate, and flexible generation of geometric diagrams for a wide array of applications.

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised learning. Additionally, unsupervised methods can provide insight into the structure of such geometrical data. At the heart of this programme is the question of how geometry can be machine learned, and indeed how AI helps one to do mathematics. This is a chapter contribution to the book Machine learning and Algebraic Geometry, edited by A. Kasprzyk et al.

Reconstructing 3D geometry from streaming video requires continuous inference under bounded resources. Recent geometric foundation models achieve impressive reconstruction quality through all-to-all attention, yet their quadratic cost confines them to short, offline sequences. Causal-attention variants such as StreamVGGT enable single-pass streaming but accumulate an ever-growing KV cache, exhausting GPU memory within hundreds of frames and precluding the long-horizon deployment that motivates streaming inference in the first place. We present OVGGT, a training-free framework that bounds both memory and compute to a fixed budget regardless of sequence length. Our approach combines Self-Selective Caching, which leverages FFN residual magnitudes to compress the KV cache while remaining fully compatible with FlashAttention, with Dynamic Anchor Protection, which shields coordinate-critical tokens from eviction to suppress geometric drift over extended trajectories. Extensive experiments on indoor, outdoor, and ultra-long sequence benchmarks demonstrate that OVGGT processes arbitrarily long videos within a constant VRAM envelope while achieving state-of-the-art 3D geometric accuracy.

Sphere packing, Hilbert's eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains unresolved: beyond a few special dimensions, neither optimal packings nor tight upper bounds are known. Even a major breakthrough in dimension n=8, later recognised with a Fields Medal, underscores its difficulty. A leading technique for upper bounds, the three-point method, reduces the problem to solving large, high-precision semidefinite programs (SDPs). Because each candidate SDP may take days to evaluate, standard data-intensive AI approaches are infeasible. We address this challenge by formulating SDP construction as a sequential decision process, the SDP game, in which a policy assembles SDP formulations from a set of admissible components. Using a sample-efficient model-based framework that combines Bayesian optimisation with Monte Carlo Tree Search, we obtain new state-of-the-art upper bounds in dimensions 4-16, showing that model-based search can advance computational progress in longstanding geometric problems. Together, these results demonstrate that sample-efficient, model-based search can make tangible progress on mathematically rigid, evaluation limited problems, pointing towards a complementary direction for AI-assisted discovery beyond large-scale LLM-driven exploration.

Despite the success of Large-Vision Language Models (LVLMs), general optimization objectives (e.g., standard MLE) fail to constrain visual trajectories, leading to language bias and hallucination. To mitigate this, current methods introduce geometric priors from visual experts as additional supervision. However, we observe that such supervision is typically suboptimal: it is biased toward geometric precision and offers limited reasoning utility. To bridge this gap, we propose Perceptual Flow Network (PFlowNet), which eschews rigid alignment with the expert priors and achieves interpretable yet more effective visual reasoning. Specifically, PFlowNet decouples perception from reasoning to establish a self-conditioned generation process. Based on this, it integrates multi-dimensional rewards with vicinal geometric shaping via variational reinforcement learning, thereby facilitating reasoning-oriented perceptual behaviors while preserving visual reliability. PFlowNet delivers a provable performance guarantee and competitive empirical results, particularly setting new SOTA records on V* Bench (90.6%) and MME-RealWorld-lite (67.0%).

We propose transferability from Large Geometric Vicinity (LGV), a new technique to increase the transferability of black-box adversarial attacks. LGV starts from a pretrained surrogate model and collects multiple weight sets from a few additional training epochs with a constant and high learning rate. LGV exploits two geometric properties that we relate to transferability. First, models that belong to a wider weight optimum are better surrogates. Second, we identify a subspace able to generate an effective surrogate ensemble among this wider optimum. Through extensive experiments, we show that LGV alone outperforms all (combinations of) four established test-time transformations by 1.8 to 59.9 percentage points. Our findings shed new light on the importance of the geometry of the weight space to explain the transferability of adversarial examples.

Recently, methods leveraging diffusion model priors to assist monocular geometric estimation (e.g., depth and normal) have gained significant attention due to their strong generalization ability. However, most existing works focus on estimating geometric properties within the camera coordinate system of individual video frames, neglecting the inherent ability of diffusion models to determine inter-frame correspondence. In this work, we demonstrate that, through appropriate design and fine-tuning, the intrinsic consistency of video generation models can be effectively harnessed for consistent geometric estimation. Specifically, we 1) select geometric attributes in the global coordinate system that share the same correspondence with video frames as the prediction targets, 2) introduce a novel and efficient conditioning method by reusing positional encodings, and 3) enhance performance through joint training on multiple geometric attributes that share the same correspondence. Our results achieve superior performance in predicting global geometric attributes in videos and can be directly applied to reconstruction tasks. Even when trained solely on static video data, our approach exhibits the potential to generalize to dynamic video scenes.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

Many geometric learning problems require invariants on heterogeneous product spaces, i.e., products of distinct spaces carrying different group actions, where standard techniques do not directly apply. We show that, when a group G acts transitively on a space M, any G-invariant function on a product space X times M can be reduced to an invariant of the isotropy subgroup H of M acting on X alone. Our approach establishes an explicit orbit equivalence (X times M)/G cong X/H, yielding a principled reduction that preserves expressivity. We apply this characterization to Equivariant Neural Fields, extending them to arbitrary group actions and homogeneous conditioning spaces, and thereby removing the major structural constraints imposed by existing methods.

Polygon representation learning is essential for diverse applications, encompassing tasks such as shape coding, building pattern classification, and geographic question answering. While recent years have seen considerable advancements in this field, much of the focus has been on single polygons, overlooking the intricate inner- and inter-polygonal relationships inherent in multipolygons. To address this gap, our study introduces a comprehensive framework specifically designed for learning representations of polygonal geometries, particularly multipolygons. Central to our approach is the incorporation of a heterogeneous visibility graph, which seamlessly integrates both inner- and inter-polygonal relationships. To enhance computational efficiency and minimize graph redundancy, we implement a heterogeneous spanning tree sampling method. Additionally, we devise a rotation-translation invariant geometric representation, ensuring broader applicability across diverse scenarios. Finally, we introduce Multipolygon-GNN, a novel model tailored to leverage the spatial and semantic heterogeneity inherent in the visibility graph. Experiments on five real-world and synthetic datasets demonstrate its ability to capture informative representations for polygonal geometries. Code and data are available at https://github.com/dyu62/PolyGNN{github.com/dyu62/PolyGNN}.

We study a version of adversarial classification where an adversary is empowered to corrupt data inputs up to some distance varepsilon, using tools from variational analysis. In particular, we describe necessary conditions associated with the optimal classifier subject to such an adversary. Using the necessary conditions, we derive a geometric evolution equation which can be used to track the change in classification boundaries as varepsilon varies. This evolution equation may be described as an uncoupled system of differential equations in one dimension, or as a mean curvature type equation in higher dimension. In one dimension, and under mild assumptions on the data distribution, we rigorously prove that one can use the initial value problem starting from varepsilon=0, which is simply the Bayes classifier, in order to solve for the global minimizer of the adversarial problem for small values of varepsilon. In higher dimensions we provide a similar result, albeit conditional to the existence of regular solutions of the initial value problem. In the process of proving our main results we obtain a result of independent interest connecting the original adversarial problem with an optimal transport problem under no assumptions on whether classes are balanced or not. Numerical examples illustrating these ideas are also presented.

Text-to-3D generation by distilling pretrained large-scale text-to-image diffusion models has shown great promise but still suffers from inconsistent 3D geometric structures (Janus problems) and severe artifacts. The aforementioned problems mainly stem from 2D diffusion models lacking 3D awareness during the lifting. In this work, we present GeoDream, a novel method that incorporates explicit generalized 3D priors with 2D diffusion priors to enhance the capability of obtaining unambiguous 3D consistent geometric structures without sacrificing diversity or fidelity. Specifically, we first utilize a multi-view diffusion model to generate posed images and then construct cost volume from the predicted image, which serves as native 3D geometric priors, ensuring spatial consistency in 3D space. Subsequently, we further propose to harness 3D geometric priors to unlock the great potential of 3D awareness in 2D diffusion priors via a disentangled design. Notably, disentangling 2D and 3D priors allows us to refine 3D geometric priors further. We justify that the refined 3D geometric priors aid in the 3D-aware capability of 2D diffusion priors, which in turn provides superior guidance for the refinement of 3D geometric priors. Our numerical and visual comparisons demonstrate that GeoDream generates more 3D consistent textured meshes with high-resolution realistic renderings (i.e., 1024 times 1024) and adheres more closely to semantic coherence.

Leveraging representation encoders for generative modeling offers a path for efficient, high-fidelity synthesis. However, standard diffusion transformers fail to converge on these representations directly. While recent work attributes this to a capacity bottleneck proposing computationally expensive width scaling of diffusion transformers we demonstrate that the failure is fundamentally geometric. We identify Geometric Interference as the root cause: standard Euclidean flow matching forces probability paths through the low-density interior of the hyperspherical feature space of representation encoders, rather than following the manifold surface. To resolve this, we propose Riemannian Flow Matching with Jacobi Regularization (RJF). By constraining the generative process to the manifold geodesics and correcting for curvature-induced error propagation, RJF enables standard Diffusion Transformer architectures to converge without width scaling. Our method RJF enables the standard DiT-B architecture (131M parameters) to converge effectively, achieving an FID of 3.37 where prior methods fail to converge. Code: https://github.com/amandpkr/RJF

Multi-view image generation holds significant application value in computer vision, particularly in domains like 3D reconstruction, virtual reality, and augmented reality. Most existing methods, which rely on extending single images, face notable computational challenges in maintaining cross-view consistency and generating high-resolution outputs. To address these issues, we propose the Geometry-guided Multi-View Diffusion Model, which incorporates mechanisms for extracting multi-view geometric information and adjusting the intensity of geometric features to generate images that are both consistent across views and rich in detail. Specifically, we design a multi-view geometry information extraction module that leverages depth maps, normal maps, and foreground segmentation masks to construct a shared geometric structure, ensuring shape and structural consistency across different views. To enhance consistency and detail restoration during generation, we develop a decoupled geometry-enhanced attention mechanism that strengthens feature focus on key geometric details, thereby improving overall image quality and detail preservation. Furthermore, we apply an adaptive learning strategy that fine-tunes the model to better capture spatial relationships and visual coherence between the generated views, ensuring realistic results. Our model also incorporates an iterative refinement process that progressively improves the output quality through multiple stages of image generation. Finally, a dynamic geometry information intensity adjustment mechanism is proposed to adaptively regulate the influence of geometric data, optimizing overall quality while ensuring the naturalness of generated images. More details can be found on the project page: https://sobeymil.github.io/GeoMVD.com.

We introduce a new method of generating Computer Aided Design (CAD) profiles via a sequence of simple geometric constructions including curve offsetting, rotations and intersections. These sequences start with geometry provided by a designer and build up the points and curves of the final profile step by step. We demonstrate that adding construction steps between the designer's input geometry and the final profile improves generation quality in a similar way to the introduction of a chain of thought in language models. Similar to the constraints in a parametric CAD model, the construction sequences reduce the degrees of freedom in the modeled shape to a small set of parameter values which can be adjusted by the designer, allowing parametric editing with the constructed geometry evaluated to floating point precision. In addition we show that applying reinforcement learning to the construction sequences gives further improvements over a wide range of metrics, including some which were not explicitly optimized.

A fundamental challenge in data science is to match disparate point sets with each other. While optimal transport efficiently minimizes point displacements under a bijectivity constraint, it is inherently sensitive to rotations. Conversely, minimizing distortions via the Gromov-Wasserstein (GW) framework addresses this limitation but introduces a non-convex, computationally demanding optimization problem. In this work, we identify a broad class of distortion penalties that reduce to a simple alignment problem within a lifted feature space. Leveraging this insight, we introduce an iterative GW solver with a linear memory footprint and quadratic (rather than cubic) time complexity. Our method is differentiable, comes with strong theoretical guarantees, and scales to hundreds of thousands of points in minutes. This efficiency unlocks a wide range of geometric applications and enables the exploration of the GW energy landscape, whose local minima encode the symmetries of the matching problem.

In recent years, 3D Gaussian splatting has emerged as a powerful technique for 3D reconstruction and generation, known for its fast and high-quality rendering capabilities. To address these shortcomings, this paper introduces a novel diffusion-based framework, GVGEN, designed to efficiently generate 3D Gaussian representations from text input. We propose two innovative techniques:(1) Structured Volumetric Representation. We first arrange disorganized 3D Gaussian points as a structured form GaussianVolume. This transformation allows the capture of intricate texture details within a volume composed of a fixed number of Gaussians. To better optimize the representation of these details, we propose a unique pruning and densifying method named the Candidate Pool Strategy, enhancing detail fidelity through selective optimization. (2) Coarse-to-fine Generation Pipeline. To simplify the generation of GaussianVolume and empower the model to generate instances with detailed 3D geometry, we propose a coarse-to-fine pipeline. It initially constructs a basic geometric structure, followed by the prediction of complete Gaussian attributes. Our framework, GVGEN, demonstrates superior performance in qualitative and quantitative assessments compared to existing 3D generation methods. Simultaneously, it maintains a fast generation speed (sim7 seconds), effectively striking a balance between quality and efficiency.

Deep neural networks are prone to adversarial examples that maliciously alter the network's outcome. Due to the increasing popularity of 3D sensors in safety-critical systems and the vast deployment of deep learning models for 3D point sets, there is a growing interest in adversarial attacks and defenses for such models. So far, the research has focused on the semantic level, namely, deep point cloud classifiers. However, point clouds are also widely used in a geometric-related form that includes encoding and reconstructing the geometry. In this work, we are the first to consider the problem of adversarial examples at a geometric level. In this setting, the question is how to craft a small change to a clean source point cloud that leads, after passing through an autoencoder model, to the reconstruction of a different target shape. Our attack is in sharp contrast to existing semantic attacks on 3D point clouds. While such works aim to modify the predicted label by a classifier, we alter the entire reconstructed geometry. Additionally, we demonstrate the robustness of our attack in the case of defense, where we show that remnant characteristics of the target shape are still present at the output after applying the defense to the adversarial input. Our code is publicly available at https://github.com/itailang/geometric_adv.

Text-to-3D generation from a single-view image is a popular but challenging task in 3D vision. Although numerous methods have been proposed, existing works still suffer from the inconsistency issues, including 1) semantic inconsistency, 2) geometric inconsistency, and 3) saturation inconsistency, resulting in distorted, overfitted, and over-saturated generations. In light of the above issues, we present Consist3D, a three-stage framework Chasing for semantic-, geometric-, and saturation-Consistent Text-to-3D generation from a single image, in which the first two stages aim to learn parameterized consistency tokens, and the last stage is for optimization. Specifically, the semantic encoding stage learns a token independent of views and estimations, promoting semantic consistency and robustness. Meanwhile, the geometric encoding stage learns another token with comprehensive geometry and reconstruction constraints under novel-view estimations, reducing overfitting and encouraging geometric consistency. Finally, the optimization stage benefits from the semantic and geometric tokens, allowing a low classifier-free guidance scale and therefore preventing oversaturation. Experimental results demonstrate that Consist3D produces more consistent, faithful, and photo-realistic 3D assets compared to previous state-of-the-art methods. Furthermore, Consist3D also allows background and object editing through text prompts.

We present a method for estimating neural scenes representations of objects given only a single image. The core of our method is the estimation of a geometric scaffold for the object and its use as a guide for the reconstruction of the underlying radiance field. Our formulation is based on a generative process that first maps a latent code to a voxelized shape, and then renders it to an image, with the object appearance being controlled by a second latent code. During inference, we optimize both the latent codes and the networks to fit a test image of a new object. The explicit disentanglement of shape and appearance allows our model to be fine-tuned given a single image. We can then render new views in a geometrically consistent manner and they represent faithfully the input object. Additionally, our method is able to generalize to images outside of the training domain (more realistic renderings and even real photographs). Finally, the inferred geometric scaffold is itself an accurate estimate of the object's 3D shape. We demonstrate in several experiments the effectiveness of our approach in both synthetic and real images.

Existing Large Multimodal Models (LMMs) struggle with mathematical geometric reasoning due to a lack of high-quality image-text paired data. Current geometric data generation approaches, which apply preset templates to generate geometric data or use Large Language Models (LLMs) to rephrase questions and answers (Q&A), unavoidably limit data accuracy and diversity. To synthesize higher-quality data, we propose a two-stage Reverse Chain-of-Thought (R-CoT) geometry problem generation pipeline. First, we introduce GeoChain to produce high-fidelity geometric images and corresponding descriptions highlighting relations among geometric elements. We then design a Reverse A&Q method that reasons step-by-step based on the descriptions and generates questions in reverse from the reasoning results. Experiments demonstrate that the proposed method brings significant and consistent improvements on multiple LMM baselines, achieving new performance records in the 2B, 7B, and 8B settings. Notably, R-CoT-8B significantly outperforms previous state-of-the-art open-source mathematical models by 16.6% on MathVista and 9.2% on GeoQA, while also surpassing the closed-source model GPT-4o by an average of 13% across both datasets. The code is available at https://github.com/dle666/R-CoT.

Large language model (LLM) agents exhibit strong mathematical problem-solving abilities and can even solve International Mathematical Olympiad (IMO) level problems with the assistance of formal proof systems. However, due to weak heuristics for auxiliary constructions, AI for geometry problem solving remains dominated by expert models such as AlphaGeometry 2, which rely heavily on large-scale data synthesis and search for both training and evaluation. In this work, we make the first attempt to build a medalist-level LLM agent for geometry and present InternGeometry. InternGeometry overcomes the heuristic limitations in geometry by iteratively proposing propositions and auxiliary constructions, verifying them with a symbolic engine, and reflecting on the engine's feedback to guide subsequent proposals. A dynamic memory mechanism enables InternGeometry to conduct more than two hundred interactions with the symbolic engine per problem. To further accelerate learning, we introduce Complexity-Boosting Reinforcement Learning (CBRL), which gradually increases the complexity of synthesized problems across training stages. Built on InternThinker-32B, InternGeometry solves 44 of 50 IMO geometry problems (2000-2024), exceeding the average gold medalist score (40.9), using only 13K training examples, just 0.004% of the data used by AlphaGeometry 2, demonstrating the potential of LLM agents on expert-level geometry tasks. InternGeometry can also propose novel auxiliary constructions for IMO problems that do not appear in human solutions. We will release the model, data, and symbolic engine to support future research.

We introduce PhysGaussian, a new method that seamlessly integrates physically grounded Newtonian dynamics within 3D Gaussians to achieve high-quality novel motion synthesis. Employing a custom Material Point Method (MPM), our approach enriches 3D Gaussian kernels with physically meaningful kinematic deformation and mechanical stress attributes, all evolved in line with continuum mechanics principles. A defining characteristic of our method is the seamless integration between physical simulation and visual rendering: both components utilize the same 3D Gaussian kernels as their discrete representations. This negates the necessity for triangle/tetrahedron meshing, marching cubes, "cage meshes," or any other geometry embedding, highlighting the principle of "what you see is what you simulate (WS^2)." Our method demonstrates exceptional versatility across a wide variety of materials--including elastic entities, metals, non-Newtonian fluids, and granular materials--showcasing its strong capabilities in creating diverse visual content with novel viewpoints and movements. Our project page is at: https://xpandora.github.io/PhysGaussian/

A recurring challenge in self-supervised learning is preventing representation collapse. Existing solutions typically rely on global regularization, such as maximizing distances, decorrelating dimensions or enforcing certain distributions. We instead reinterpret representation learning as a discrete packing problem, where preserving information simplifies to maintaining injectivity. We operationalize this in Hypersolid, a method using short-range hard-ball repulsion to prevent local collisions. This constraint results in a high-separation geometric regime that preserves augmentation diversity, excelling on fine-grained and low-resolution classification tasks.

When learning simulations for modeling physical phenomena in industrial designs, geometrical variabilities are of prime interest. While classical regression techniques prove effective for parameterized geometries, practical scenarios often involve the absence of shape parametrization during the inference stage, leaving us with only mesh discretizations as available data. Learning simulations from such mesh-based representations poses significant challenges, with recent advances relying heavily on deep graph neural networks to overcome the limitations of conventional machine learning approaches. Despite their promising results, graph neural networks exhibit certain drawbacks, including their dependency on extensive datasets and limitations in providing built-in predictive uncertainties or handling large meshes. In this work, we propose a machine learning method that do not rely on graph neural networks. Complex geometrical shapes and variations with fixed topology are dealt with using well-known mesh morphing onto a common support, combined with classical dimensionality reduction techniques and Gaussian processes. The proposed methodology can easily deal with large meshes without the need for explicit shape parameterization and provides crucial predictive uncertainties, which are essential for informed decision-making. In the considered numerical experiments, the proposed method is competitive with respect to existing graph neural networks, regarding training efficiency and accuracy of the predictions.