# NPR: Nocturnal Place Recognition in Streets

Bingxi Liu1,2, Yujie Fu3, Feng Lu2,4, Jinqiang Cui2, Yihong Wu3, Hong Zhang1,∗
1Southern University of Science and Technology, Shenzhen, China.
2Peng Cheng Laboratory, Shenzhen, China.
3Institute of Automation, Chinese Academy of Sciences, Beijing, China.
4Tsinghua University, Shenzhen, China.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Visual Place Recognition (VPR) is the task of retrieving database images
similar to a query photo by comparing it to a large database of known images.
In real-world applications, extreme illumination changes caused by query
images taken at night pose a significant obstacle that VPR needs to overcome.
However, a training set with day-night correspondence for city-scale, street-
level VPR does not exist. To address this challenge, we propose a novel
pipeline that divides VPR and conquers Nocturnal Place Recognition (NPR).
Specifically, we first established a street-level day-night dataset,
NightStreet, and used it to train an unpaired image-to-image translation
model. Then we used this model to process existing large-scale VPR datasets to
generate the VPR-Night datasets and demonstrated how to combine them with two
popular VPR pipelines. Finally, we proposed a divide-and-conquer VPR framework
and provided explanations at the theoretical, experimental, and application
levels. Under our framework, previous methods can significantly improve
performance on two public datasets, including the top-ranked method.

##  1 Introduction

Visual Place Recognition (VPR) is a fundamental task in the fields of computer
vision [4, 19, 34, 37, 38, 48] and robotics [16, 22, 23, 25, 28, 43], which
involves returning database images that are similar to a query image by
comparing it with a known large-scale database image. As previously reported
in research, challenges for VPR include database scale [6], repeated
structures [38], structural changes [18], occlusion [4], viewpoint[7], visual
scale [14], illumination [3, 9, 23, 22, 32, 36, 37, 43, 48], and seasonal
changes [25, 35]. Almost all recent VPR methods are learning on large-scale
datasets [6, 24, 26, 37, 45]. A neural network is often used to map images
into an embedding space that can efficiently distinguish images captured from
different places. However, existing datasets restrict the progress of VPR in
nighttime street scenes.

Figure 1: A demo for NPR. A 1.2km $\times$ 1.2km satellite map is presented,
where densely purple dots represent the locations where the database images
were captured. Our proposed method can retrieve the correct results from a
database of 75,984 images, even when provided with a nighttime captured image
that includes a significant change in the view direction.

Training sets. Previous research has established large-scale VPR training sets
using the downloading interface of Google Street View [1, 2, 4, 6]. These
training sets were collected using either car-mounted panoramic cameras or
pedestrian-carried street-view cameras and covered almost all of the
challenges above, except for nighttime scenes. Some VPR training sets for
autonomous driving scenes include nighttime scenes but lack other challenges
[23, 43]. For example, cars are limited to roads and mostly use forward-facing
cameras, resulting in a lack of view direction changes.

Testing sets. Two representative testing sets for VPR in nighttime street
scenes are Tokyo 24/7 [37] and Aachen Day/Night. i) The Tokyo 24/7 dataset
comprises daytime, sunset, and nighttime scenes. However, all research [4, 6,
8, 15, 21] has yet to separate these scenes for testing purposes; even the
original paper [37] tested only sunset and nighttime together. This oversight
obscures the fact that VPR performance is poor at night. ii) The Aachen
Day/Night dataset evaluates Visual Localization (VL) but can also assess VPR
as it serves as the first stage in a two-stage VL approach [12, 30, 33]. The
dataset is divided into daytime and nighttime and ranked on an evaluation
system without visible ground truth. The top-ranked method used VPR to recall
50 candidate frames, which is poor in practical applications.

Under the limitations of the training set, a straightforward idea is to apply
Image Enhancement (IE) or image-to-image translation for the nighttime queries
to daytime queries. However, these methods or their application in VPR have
the following problems: i) These methods introduce additional computing
resources and time. ii) Learning-based IE’s training sets usually pairs of
low-exposure to high-exposure images rather than night-to-day pairs [44], even
not outdoors, so they may have poor performance on the VPR dataset. iii)
Inaccurate or erroneous night-to-day image-to-image translation can result in
a degradation of VPR performance [3].

In this article, we successfully address training set issues through reverse
”Night-to-Day” and propose a method to divide VPR and conquer Nocturnal Place
Recognition (NPR). To summarize, our contributions are as follows:

  * •

We propose a dataset comprising street scene images captured during daytime
and nighttime and trained an unpaired image-to-image translation network on
this dataset.

  * •

Using the above translation network, we processed existing VPR datasets to
generate the VPR-Night datasets and demonstrated two popular VPR pipelines on
how to leverage the VPR-Night datasets.

  * •

We propose a divide-and-conquer VPR framework and provide theoretical,
experimental, and practical explanations for the framework. Furthermore, under
this framework, previous methods exhibit superior performance on public
datasets.

##  2 Related Work

Visual Place Recognition has been dominated by deep learning methods. Previous
research can be summarized in three main aspects: model, loss function, and
data. At the model level, Convolutional Neural Networks (CNNs) [4] or Visual
Transformers (ViTs) [41] are typically used as the feature extraction network
backbone, followed by a pooling or aggregation layer [4, 29], such as NetVLAD.
At the loss function level, triplet loss and contrastive loss are commonly
used to increase the Euclidean margin for better feature embedding [8].
However, triplet loss has a significant issue with mining hard negative
samples. To address this problem, Liu et al. [21] proposed the statistical
attention embedding loss, which efficiently utilizes multiple negative samples
for supervision. Moreover, building on [15], Ge et al. [15] utilized network
output scores as self-supervised labels and achieved new state-of-the-art
results through iterative training. Berton et al. [6] used the Large Margin
Cosine Loss (LMCL) in VPR tasks and demonstrated superior performance over
triplet loss. At the data level, it can be further divided into two scenarios:
road scenes and street scenes. Road scene datasets typically have obvious
characteristics [16, 23, 43], such as the camera facing forward and the fixed
viewing direction. Although these datasets may contain nighttime data, these
characteristics do not suit VPR tasks in street scenes. Street scene datasets
are usually obtained from the Google Street View download interface [1, 2, 6],
which can be arbitrarily expanded and includes almost all challenges of VPR
tasks, except for nighttime scenes. Therefore, it is reasonable to conclude
that the performance of existing VPR models in nighttime scenarios benefits
from data augmentation during the training phase and the generalization
capability during the inference phase.

Nighttime Computer Vision involves addressing classic downstream tasks using
nighttime images, which can be categorized into two-stage and one-stage
methods. Two-stage methods [3, 46] involve converting nighttime images to
daytime images before performing downstream tasks, while one-stage
methodologies exploit raw nighttime data for training such tasks [11, 48]. For
instance, Anoosheh et al. [3] propose a GAN method for converting nighttime
images into daytime images and perform retrieval-based localization. Cui et
al. [11] propose a method to perform reverse ISP and dark processing and train
an end-to-end model for dark object detection. Xu et al. [46] propose a GAN-
based approach to convert nighttime images into daytime images for semantic
segmentation of autonomous driving scenes. However, in the context of street-
level VPR, two-stage methods suffer from poor generalization and increased
computational requirements, whereas the one-stage approach is constrained by a
scarcity of suitable training datasets. To address this limitation, we
introduce a novel paradigm that integrates the two perspectives by
transforming daytime images into nighttime images prior to VPR training.

Image-to-Image Translation is a class of vision and graphics problems where
the goal is to learn the mapping between an input image and an output image
using a training set. The classification of these methods is based on the type
of training set, which can be pixel-level corresponding image-to-image or
unpaired image-to-image [50, 47]. Obtaining large-scale, street-level, pixel-
level correspondence day-night image pairs in the real world is an extremely
challenging task. Therefore, we have considered the second type of training
set. There are two main categories of methods used for unpaired image-to-image
translation: two-side mapping and one-side mapping. The former, which includes
CycleGAN [50] and DualGAN [47], relies on the cycle-consistency constraint.
This constraint ensures that the translated image can be reconstructed using
an inverse mapping. However, the bijective projection required by this
approach is often too limiting, which has led to the development of one-side
mapping techniques. One such approach involves using a geometry distance to
preserve content. DistanceGAN [13], for instance, maintains the distances
between images within domains, while GCGAN [5] ensures geometry-consistency
between input and output. Another technique, CUT [27], maximizes mutual
information between the input and output using contrastive learning. In
particular, the diffusion model [49] has recently demonstrated outstanding
performance in solving image-to-image translation problems. However, its usage
necessitates substantial computational resources. After a thorough comparison
of several approaches, we proceeded to train our day-to-night image-to-image
translation network utilizing NEGCUT [42] as the foundation.

##  3 The Proposed Dataset

In this chapter, we introduce a series of datasets that have been proposed,
which are divided into two categories: NightStreet dataset for training a day-
to-night image-to-image translation network and VPR-Night dataset for training
night-to-day VPR networks.

###  3.1 The NightStreet Dataset

Figure 2:  Examples of Day-Night Style Datasets. (a) is the LOL dataset [44],
(b) is the Robotcar dataset [23], (c) is the Night2Day dataset [20], and (d)
is the NightStreet dataset.

Some datasets have been previously identified as potentially useful for day-
to-night image-to-image translation. Nevertheless, they are not suitable for
VPR tasks in street scenes. As depicted in Figure 2 (a), the current low light
enhancement research predominantly focuses on tackling low light conditions
(such as those caused by backlighting) or weak exposure. However, utilizing
these datasets in reverse fails to accurately capture the changes that occur
during day-to-night translation. As demonstrated in Figure 2 (b), there exist
significant differences between automatic driving scenes and street scenes.
For example, roadways typically constitute more than one-third of the image
content, and images captured while the car is in motion tend to exhibit
blurriness. Additionally, lighting conditions (e.g., street lights and car
tail lights) generally exhibit limited variation. In regards to Figure 2 (c),
several time-lapse photography datasets have been proposed in the image
generation field. However, these datasets share a common characteristic: the
photographers who capture them tend to focus on distant views and skylines,
which differ greatly from the urban street scenes.

We constructed day-night image pairs by directly rearranging the query images
from Tokyo 24/7 and Aachen Day/Night datasets. Tokyo 24/7 [37] provides 375
daytime and nighttime images each, while Aachen Day/Night [33] includes 234
daytime and 196 nighttime images. Importantly, we did not exploit the ground-
truth relationship between the query and database images. Instead, we trained
our translation network under unpaired images setting.

###  3.2 The VPR-Night Datasets

We used a trained day-to-night image-to-image translation model to process the
existing VPR datasets and obtained the VPR-Night datasets. Theoretically, this
method can be applied to any VPR dataset. In this study, we processed the
Pitts-30k and SF-XL-small datasets, and named the new datasets Pitts-30k-N and
SF-XL-small-N, respectively.

Pitts-30k-N. Pitts-30k is a subset of Pittsburgh-250k that is widely used in
the research of VPR because of its high experimental efficiency [4]. It
consists of 30k database images from Google Street View and 24k test queries
generated from Street View but taken at different times, and is divided into
three roughly equal parts for training, validation, and testing. According to
the method designed in section 4.2, we only need to perform day-to-night
transfer on the test queries. Therefore, there are 24k night-style query
images and 30k daytime database images in Pitts-30k-N dataset.

SF-XL-small-N. The San Francisco eXtra Large (SF-XL) [6] is a city-scale,
dense, time-varying dataset. It crawls 3.43M equirectangular panoramas images
from Google Street View and divides them into 12 crops, with the entire
dataset consisting of 41.2M images. Each crop is labeled with 6 DoF
information including GPS and heading orientation. Unlike the Pittsburgh
dataset, the training set is not divided into query images and database
images. To quickly validate our method, we opted to use a subset of SF-XL,
namely SF-XL-small, comprising 100k street view images. Similarly, SF-XL-
small-N contains 100k original images and 100k nighttime images.

Figure 3: Schematic Diagrams of Two Popular VPR Pipelines. In (a), the blue,
green, and red blocks inside the sphere represent anchor, positive, and
negative samples in the embedding space. In (b), the blue, green, and red
blocks inside the sphere belong to different classes in the embedding space.
In (a) and (b), the black blocks represent nighttime images.

##  4 Method

In this chapter, we first introduce the image-to-image translation from day-
to-night, and then describe how to train two VPR pipelines on the generated
nighttime data. Finally, we explain the rationale and implementation of
dividing VPR and conquering NPR.

###  4.1 Day-to-Night Image-to-Image Translation

In this section, we introduce a contrastive learning-based unpaired image-to-
image translation network [27, 42], which is trained on the NightStreet
dataset and generates the VPR-Night datasets. Our goal is to preserve the
content of daytime images while specifying nighttime style. We define the
daytime images from the input domain as $\mathcal{X}\subset\mathbb{R}^{H\times
W\times 3}$ and the nighttime images from the output domain as
$\mathcal{Y}\subset\mathbb{R}^{H\times W\times 3}$, and aim to learn a mapping
$G:\mathcal{X}\to\mathcal{Y}$. The NightStreet dataset can be represented as
an instance $X=\\{x\in\mathcal{X}\\}$ and $Y=\\{y\in\mathcal{Y}\\}$. The
mapping function $G$ is decomposed into an encoder $G_{\mathrm{enc}}$ and a
generator $G_{\mathrm{dec}}$, so the process of producing output images can be
represented as:

$\hat{y}=G(x)=G_{\mathrm{dec}}(G_{\mathrm{enc}}(x)).$ (1)

We encourage the output images to have a nighttime style similar to the target
domain by using a discriminator $D$ and the following adversarial loss [50]:

$\mathcal{L}_{\mathrm{GAN}}=\mathbb{E}_{y\sim Y}\log{D(y)}+\mathbb{E}_{x\sim
X}\log{(1-D(G(x)))}.$ (2)

Then, a contrastive learning framework is employed to maintain local content
consistency between input $x$ and output $\hat{y}$. We extract a certain
number $N$ of patches from the images, where one pair of patches is located at
the same position in $x$ and $\hat{y}$, denoted as $q$ and $p$, while the
remaining $N-2$ patches are randomly selected from the $x$. These patches are
then fed into a feature extraction network to obtain feature vectors, and an
$(N-1)$ classification problem is established. The feature extraction network
used here is based on the $G_{\mathrm{enc}}$, followed by a 2-layer MLP.

###  4.2 Two Night-to-Day VPR Pipelines

In this section, we introduce strategies for utilizing nighttime data to train
two popular VPR frameworks, namely the triplet network shown in Figure 3 (a)
and the classification network shown in Figure 3 (b).

Triplet Network. From the training set with GPS, anchor samples $q$ are
selected, and data sharing the same GPS label or with close proximity are
considered positive samples $p$, while the remaining samples are regarded as
negative samples $n$. These samples are then fed into $f$, a pre-trained deep
neural network with an aggregation layer, obtaining feature vectors
$f(q),f(p),$ and $f(n)$ in the embedding space. Finally, the Euclidean
distance between the $f(q)$ and $f(p)$, as well as $f(q)$ and $f(n)$, are
calculated in the embedding space to obtain a triplet loss, which is
formulated as:

$L_{\mathrm{triplet}}=l(||f(q)-f(p)||_{2}^{2}-||f(q)-f(n)||_{2}^{2}+m),$ (3)

where $l$ is the hingle loss $l(x)=\mathrm{max}(x,0)$, $m$ is the margin
parameter that controls the distance between samples of the same class and
different classes.

Considering that we aim to achieve Night-to-Day VPR, we need to construct
sample pairs with different styles, i.e., we need to transfer the anchor
samples to nighttime style or convert both positive and negative samples to
nighttime style. The latter method is obviously less efficient than the
former.

Classification Network. VPR can be treated as a classification problem based
on labels. Following [6], the training set can be partitioned into classes by
splitting it into square geographical cells using UTM coordinates
$\\{\textit{east},\textit{north}\\}$ and further dividing each cell into a set
of classes based on the orientation$\\{\textit{heading}\\}$ of each image. We
transformed all images in the database into a night style while preserving
their original labels. Then we employed the Large Margin Cosine Loss (LMCL)
[40] to train a model:

$L_{\mathrm{lmc}}=\frac{1}{N}\sum_{i}{-\log{\frac{e^{s(\cos(\theta_{{y_{i}},i})-m)}}{e^{s(\cos(\theta_{{y_{i}},i})-m)}+\sum_{j\neq
y_{i}}{e^{s\cos(\theta_{j,i})}}}}},$ (4)

subject to

$\begin{split}cos(\theta_{j},i)&={W_{j}}^{T}x_{i},\end{split}$ (5)

where $N$ is the numer of training images, $x_{i}$ is the $i$-th embedding
vector corresponding to the ground truth class of $y_{i}$, the $W_{j}$ is the
weight vector of the $j$-th class, and $\theta_{j}$ is the angle between
$W_{j}$ and $x_{i}$. $s$ and $m$ are two hyperparameters that control the
weights of the intra-class distance and inter-class distance in the loss
function, respectively.

###  4.3 Dividing VPR and Conquering NPR

In this section, we introduce concepts from three fields: deep learning,
neuroscience, and computer algorithms. Our aim is to explain the rationale
behind the need to divide night vision problem from general visual
problem111Although we focus solely on the VPR task, we believe that this
framework should be applicable to many other computer vision tasks..

Deep learning. We utilize data-driven network learning, where the training set
and test set should have similar distributions [17]. However, previous
research on nighttime VPR does not follow this principle, which is the reason
why all methods degrade severely in nighttime scenes. When the fitting ability
of the model is limited, increasing the amount of nighttime data will also
cause the VPR performance in daytime scenes to degrade.

Neuroscience. There are two types of photoreceptor cells distributed on the
retina: cone cells and rod cells [39]. Cone cells are primarily responsible
for color and detail discrimination and are only activated under relatively
adequate lighting conditions. Rod cells, on the other hand, are mainly
responsible for identifying the intensity of light and motion, and can be
responsive under low-light conditions. Nocturnal animals possess a greater
number of rod cells.

Computer algorithm. Divide-and-conquer [10] (D&C) is a classic algorithm that
decomposes a large problem into several smaller but structurally similar sub-
problems, recursively solving these sub-problems, and then combining the sub-
problem solutions, to obtain the original problem solution.

Based on the above knowledge, it is suggested that the original model should
be used for daytime scenes, while the VPR-Night trained or fine-tuned model
should be used for nighttime scenes. Then we provide three implementation
ideas for the ”divide” step: i) training a day-night network for
classification, ii) using the global average brightness and an empirical
threshold for classification, and iii) using system time and local sunset time
222https://www.timeanddate.com/astronomy/ for classification. The first method
can achieve better results, but has low practicality in real-world
applications. The second method may fail in some specific situations, such as
scenes with strong backlight being misclassified as night and scenes with
bright light being misclassified as day. The third method is independent of
image content, but is only applicable to devices equipped with communication
functions, such as smartphones and robots. In our experiments, we use a
combination of the second and third methods for decomposing the test set.
Finally, we emphasize that the execution of ”divide” in the real world is a
one-time process, so the additional computation required for new pipelines can
be ignored.

##  5 Experiments

Method | Backbone |  | Aggregation
---
Method
| Feature
---
Dim
| Loss
---
Function
| Training
---
Dataset
| R@1
---
All queries
| R@1
---
Day queries
| R@1
---
Sunset queries
| R@1
---
Night queries
| NetVLAD [4] (CVPR’16)
---
NetVLAD-NPR
NetVLAD-D&C
| VGG-16
---
| NetVLAD
---
| 32768
---
| Triplet Loss
---
| Pitts-30k
---
Pitts-30k-N
-
| 63.8
---
62.5 (-1.3)
68.6 (+4.8)
| 88.6
---
80.0 (-8.6)
88.6
| 75.2
---
64.8 (-10.4)
74.3 (-0.9)
| 27.6
---
42.9 (+15.3)
42.9 (+15.3)
NetVLAD [4] (CVPR’16) | VGG-16 | NetVLAD+PCA | 4096 | Triplet Loss | Pitts-30k | 68.9 | - | - | -
DIR [29] (T-PAMI’18) | Res-101 | GeM+FC | 2048 | Triplet Loss | Google Landmark | 74.9 | 92.4 | 81.9 | 50.5
SARE [21] (ICCV’19) | VGG-16 | NetVLAD+PCA | 4096 | SARE-Joint | Pitts-30k | 74.8 | - | - | -
SFRS [15] (ECCV’20) | VGG-16 | NetVLAD+PCA | 4096 | SARE-Joint | Pitts-30k | 78.5 | - | - | -
APPSVR [28] (ICCV’21) | VGG-16 | APP+PCA | 4096 | Triplet Loss | Pitts-30k | 77.1 | - | - | -
| DVG [8] (CVPR’22)
---
DVG-NPR
DVG-D&C
| Res-18
---
| NetVLAD
---
| 16384
---
| Triplet Loss
---
| Pitts-30k
---
Pitts-30k-N
-
| 66.7
---
74.0 (+7.3)
74.9 (+8.2)
| 90.5
---
85.7 (-4.8)
90.5
| 76.2
---
80.0 (+3.8)
78.1 (+1.9)
| 33.3
---
56.2 (+22.9)
56.2 (+22.9)
| CosPlace [6] (CVPR’22)
---
CosPlace-NPR
CosPlace-D&C
| VGG-16
---
| GeM+FC
---
| 512
---
| LCM Loss
---
| SF-XL
---
Fine tuning on small-N
-
| 81.9
---
82.9 (+1.0)
84.1 (+2.2)
| 90.5
---
87.6 (-2.9)
90.5
| 89.5
---
83.8 (-5.7)
84.8 (-4.7)
| 65.7
---
77.1 (+11.4)
77.1 (+11.4)
| CosPlace [6]
---
CosPlace-NPR
| Res-50
---
| GeM+FC
---
| 512
---
| LCM Loss
---
| SF-XL
---
Fine tuning on small-N
| 88.6
---
92.1 (+3.5)
| 95.2
---
96.2 (+1.0)
| 90.5
---
93.3 (+2.8)
| 80.0
---
86.7 (+6.7)

Table 1: Comparisons of various methods on Tokyo 24/7 [37]. We reproduced
three methods and added their NPR and D&C versions. As previously mentioned,
none of the previous methods conducted experiments on the last three columns.
Therefore, we currently only report their Recall@1 metrics for all queries.

In this chapter, we describe the implementation details, the test set, and the
evaluation methods. We provide quantitative and qualitative results, followed
by visualizations that demonstrate the utility of NPR as well as its current
limitations.

###  5.1 Implementation details

Day-to-Night Image-to-Image Translation. We use the same training parameters
as in [42] and select the model from the 400th epoch for inference. Since the
image size of NightStreet is complex and differs greatly from that of the VPR
training set, we scale the NightStreet images proportionally with the shorter
side fixed at 640 and then randomly crop them to a size of 512$\times$512
during training. This can maintain scale consistency between the training set
and testing set as much as possible. The translation network takes
approximately one day to process the SF-XL-small dataset, at a resolution of
512$\times$512 on a single 3090 Ti.

Night-to-Day Visual Place Recognition. The essence of our method is to
transfer the nighttime style to existing datasets and combine it with existing
pipelines. Therefore, it can be compatible with any VPR method and improve
their performance in nighttime scenes. We replicated the work of [4, 8, 6],
and applied our method. Since most of the previous methods used the VGG-16
structure, we trained and compared it with this structure. The current top-
ranked method on Tokyo 24/7 is based on ResNet-50, so we also conducted
experiments based on it. We reproduce the best performance of the baseline
method or directly use the model open-sourced by the author. Specifically,
data augmentation was not used during NPR training and the learning rate used
when fine-tuning the model was 1e-6.

Night-to-Day Visual Localization. We adopted the hierarchical localization
framework provided by [30] and replaced the VPR module with Superpoint [31]
and Superglue [12] for local feature extraction and matching, respectively.
Notably, we adjusted the input image resolution of the VPR module to the
recommended parameters for this project.

###  5.2 Datasets and evaluation methodology

We reported results on multiple publicly available datasets.

Tokyo 24/7 v2 [37] contains 75,984 database images from Google Street View and
315 query images taken using mobile phone cameras. This is an extremely
challenging dataset where the queries were taken at daytime, sunset,
nighttime, while the database images were only taken at daytime. However, to
the best of our knowledge, all works using this dataset have not tested them
by category. Even the original work that proposed this dataset evaluated
sunset and night as one category. This incident seriously overshadowed the
fact that the performance of VPR was poor under the cover of night. We propose
two ways to test the dataset: one is to directly split by labels, and the
other is to use the information of exchangeable image file format (EXIF) and
local sunset time
333https://www.timeanddate.com/sun/japan/tokyo?month=9&year=2014 for
partitioning. The second method will split the sunset testing set into two
categories, test them separately, and then merge the results.

Aachen Day/Night v1 [33] comprises 922 query images captured during day and
night, and 4328 reference images collected over a span of two years using
hand-held cameras. All query images were captured using mobile phone cameras,
hence the Aachen Day-Night dataset considers the scenario of localization
using mobile devices, e.g., for augmented or mixed reality applications.

Other datasets. Additionally, we employ the Pitts-30k and SF-XL-test-v1
datasets to investigate the phenomenon of model degradation.

Evaluation metric. In the VPR dataset, we use the recall@N with a 25 meters
threshold, i.e., the percentage of queries for which at least one of the first
N predictions is within a 25 meters distance from the ground truth position of
the query, following standard procedure [4, 37, 28, 6, 21, 15]. In the visual
localization dataset, we evaluate VPR using the success rate of localization
under different recall@N metrics. Specifically, the visual localization system
considers higher localization precision than that required by the VPR dataset.

###  5.3 Comparison with the State-of-the-art Methods

Figure 4: Comparison of our methods and baseline. The solid line represents
our method, while the dashed line represents the baseline method. It should be
noted that the y-axis origins of the two graphs are different.

Method | Backbone |  Aggregation Method |  Feature Dim | (0.25m, 2°) / (0.5m, 5$\degree$) / (5m, 10°)
---|---|---|---|---
R@1 | R@5 | R@10 | R@20
| NetVLAD [4]
---
VGG-16 | NetVLAD+PCA | 4096 |  | 68.4 / 78.6 / 85.7
---
| 75.5 / 84.7 / 93.9
---
|  82.7 / 88.8 / 95.9
---
| 83.7 / 92.9 / 99.0
---
| CosPlace
---
CosPlace-NPR
Res-50 | GeM | 512 |  | 69.4 / 82.7 / 89.8
---
71.4 / 84.7 / 93.9
| 77.6 / 87.8 / 95.9
---
78.6 / 88.8 / 95.9
| 77.6 / 87.8 / 95.9
---
79.6 / 88.8 / 95.9
| 83.7 / 90.8 / 98.0
---
84.7 / 91.8 / 98.0

Table 2: Comparisons of three methods on Aachen Day/Night [36]. All three
methods employ the same VL pipeline [30], with only the VPR method being
replaced. The success rates under different levels of localization accuracy
were evaluated at Recall@N.

As shown in Table 1, we have replicated the experimental results of NetVLAD
[4], DIR [29], DVG [8], and CosPlace [6] on the Tokyo 24/7 dataset, while
citing the results of NetVLAD with PCA [4], SARE [21], SFRS [15], and APPSVR
[28] from the Deep Visual Geo-Localization Benchmark [8]. Our method was
applied to these three replication methods and labeled with the name suffixes
NPR and D&C accordingly. The results can be summarized in a few points:

1) The Recall@1 for nighttime queries are significantly lower than those for
daytime queries across all methods. The Recall@1 for all queries is a trade-
off result. This confirms our viewpoint that the challenge of Nighttime VPR
has been overlooked for a considerable period of time.

2) All methods trained on the VPR-Night datasets showed a significant
improvement in performance on nighttime queries. Models with weaker fitting
abilities, such as VGG-16 and ResNet-18, showed a corresponding degradation in
daytime scenes, whereas networks with stronger fitting abilities, such as
ResNet-50, were able to improve Recall@1 for both daytime and nighttime
queries.

3) To address the issue of imbalanced performance between daytime and
nighttime for small models, a divide-and-conquer algorithm can effectively
maintain performance balance, which is particularly beneficial for models
deployed on mobile platforms.

As illustrated in Figure 4, we present the variation of accuracy with respect
to different recall values, N. Our proposed method shows a significant
performance improvement over the baseline approach across all recall values.
It is worth noting that the vertical axes of the two plots have different
origins.

As shown in Table 2, our proposed method outperforms the baseline approach on
the nighttime testing subset of the Aachen dataset. While our localization
success rates at R@10 and R@20 are slightly lower than those of NetVLAD with
PCA, we think this may be attributed to the fact that the size of the database
cannot fully reflect the advantages of our model, and our output
dimensionality is also significantly lower than that of NetVLAD.

###  5.4 Daytime VPR experiments

| Pitts-30k-test | SF-XL-test-v1
---|---|---
| R@1 | R@5 | R@1 | R@5
CosPlace | 88.4 | 94.6 | 68.1 | 78.9
CosPlace-NPR | 88.0 (-0.4) | 94.1 (-0.5) | 65.4 (-2.7) | 75.8 (-3.1)

Table 3: Results on Pitts-30k-test [4] and SF-XL-test-v1 [6]. Our Model is
CosPlace+NPR (VGG-16), while the comparative method is CosPlace (VGG-16). It
should be noted that for testing SF-XL-test-v1, we used the SF-XL-small
database instead of SF-XL. Therefore, the CosPlace’s result presented here are
slightly better than that reported in [6].

While we did not have high expectations for the performance of the NPR during
daytime, we still demonstrated its performance on other daytime datasets and
compared it with baseline methods. As shown in Table 3, our method exhibits a
slight decrease in performance on the Pitts-30k and SF-XL-test-v1 datasets.

###  5.5 Qualitative experiment

Figure 5: Examples of translation results from SF-XL-small-N. Each row
corresponds to one location. (a), (c), and (e) represent the images captured
at different times for the same location, while (b), (d), and (f) correspond
to the nighttime images generated from (a), (c), and (e), respectively.
Figure 6: Examples of retrieval results for challenging queries on Tokyo 24/7.
Each column corresponds to one query case: the query is shown in the first
row, the top retrieved image using our best method (CosPlace-NPR) in the
second, and the top retrieved image using our best baseline (CosPlace) in the
last row. Correct retrievals are indicated with a green border, while
incorrect retrievals are indicated with a red border.

In this section, we present samples from the generated VPR-Night datasets and
the results of CosPlace-NPR. As shown in Figure 5, (1) and (2) represent two
locations, (a), (c), and (e) are data from different times of day in Google
Street View, while (b), (d), and (f) are generated results. Our method
preserves scale, appearance, and viewpoint changes while adding a nighttime
style.

As shown in Figure 6, our incremental improvement significantly enhances
performance at night. These nocturnal query images not only exhibit extreme
illumination differences with database images, but also undergo architectural
resets, changes in perspective, and scale variations. This is highly
consistent with real-world scenarios.

###  5.6 Limitations

We believe that our theory is not limited, but the current implementation is
constrained. i) We require a larger NightStreet dataset to capture a richer
range of day-to-night variations. Fortunately, the loose requirement for
unpaired images makes the dataset easily extensible. And constructing the
NightStreet dataset is undoubtedly less challenging than creating a large-
scale, street-level, day-to-night corresponding VPR dataset. ii) Rendering
large-scale datasets such as SF-XL requires significant GPU resources. We plan
to address both of these limitations in our future work.

##  6 Conclusions

In this work, we address the challenging problem of nighttime VPR, which has
been hindered by the lack of appropriate training datasets and inaccurate
testing methodologies. To overcome these issues, we propose a dedicated
pipeline for Noctural Place Recognition. First, we construct the NightStreet
dataset and train a day-to-night image-to-image translation network. We then
apply the network to process existing large-scale VPR datasets and demonstrate
how to integrate them into two popular VPR pipelines. Finally, we introduce
the idea of differentiating between VPR and NPR, providing a multidimensional
interpretation. Our experimental results show that our pipeline significantly
improves previous methods.

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#  A First Analysis of Stochastic Composite Gravity

Joshua Erlich<EMAIL_ADDRESS>High Energy Theory Group, Department of
Physics, William & Mary, Williamsburg, VA 23187-8795, USA

(August 18, 2022)

###### Abstract

We present a first analysis of a nonperturbative approach to quantum gravity
based on a representation of quantum field theory in terms of stochastic
processes. The stochastic description accommodates a physical Lorentz-
invariant ultraviolet regulator that provides a novel description of physics
at ultra-short distances. In a stochastic composite gravity model we
demonstrate the evolution of a generic initial field configuration towards an
equilibrium in which the composite spacetime metric fluctuates about a flat
spacetime. We argue that fluctuations about the vacuum give rise to an
emergent gravitational interaction consistent with Einstein gravity at long
distances. We uncover a formal similarity between regularization by stochastic
discreteness and point-splitting regularization in the corresponding quantum
field theory. We comment on the signature of the emergent spacetime, possible
consequences for the early universe, and the potential for observational and
experimental tests of the stochastic origin of quantum field theory and
gravitation.

##  I Introduction

The development of a quantum theory of gravity has been hindered by a number
of conceptual and technical challenges stemming from diffeomorphism
invariance, spacetimes with nontrivial causal structure, and the role of time
as a parameter for the dynamics. A clue was provided by Sakharov’s observation
that diffeomorphism invariance should lead to the appearance of the Einstein-
Hilbert action in the partition function of a generic regularized quantum
field theory in a background spacetime [1, 2, 3]. However, it has remained
unclear what benefit might be drawn from this observation because the
spacetime degrees of freedom would still require quantization.

Related to Sakharov’s observation is the paradigm of composite gravity,
according to which there is no fundamental spacetime geometry, but instead the
role of the vielbein or metric is played by a composite operator composed of
the matter and gauge fields [4, 6, 5]. In composite gravity the gravitational
interaction arises as an artifact of the ultraviolet regulator, so the
regulator cannot be removed in a limit. Hence, identifying a physical
regulator is critical to the success of the composite gravity program. A
number of analyses have confirmed the emergence of a gravitational interaction
in toy models of composite gravity, albeit with traditional unphysical
regulators such as dimensional regularization and Pauli-Villars fields which
are generally believed to lead to inconsistencies as the regulator scale is
approached [7, 9, 8, 10, 11, 12].

In this paper we present a first analysis of a framework for quantum gravity
based on a representation of quantum field theory in terms of stochastic
processes. Discreteness of random fluctuations of the matter fields provides a
physical ultraviolet regulator and a novel description of physics at
ultrashort distances from which both quantum field theory and gravitation
emerge as effective descriptions in an appropriate regime. The suggestion that
quantum gravity might arise from a definition of quantum field theory in terms
of stochastic processes is not novel [14, 15, 16, 13, 17, 18], but to the
author’s knowledge this is the first concrete realization of the paradigm in a
composite gravity setting.

##  II Composite Gravity

There is a straightforward algorithm, at least in principle, for constructing
a composite gravity model whose effective description at long distances and
weak fields is a specified quantum field theory coupled to gravity: We begin
with the field theory coupled to an auxiliary spacetime background as in
Sakharov’s induced gravity scenario, and we replace the spacetime metric or
vielbein (up to local Lorentz transformations) with the solution to the
constraint $T_{\mu\nu}=0$, where $T_{\mu\nu}$ is the energy-momentum tensor in
the curved-space background. Here we consider the scalar toy model
investigated in Refs. [6, 7], which may be obtained from a theory of $N\gg 1$
real scalar fields minimally coupled to an auxiliary metric $g_{\mu\nu}$ with
action,

$S_{{\rm aux}}=\int
d^{d}x\sqrt{g}\left\\{\frac{1}{2}\sum_{a=1}^{N}\partial_{\mu}\phi^{a}\,\partial_{\nu}\phi^{a}\,g^{\mu\nu}-V(\phi^{a})\right\\}.$
(1)

The constraint of vanishing energy-momentum tensor determines the auxiliary
metric as a function of the fields:

$\displaystyle
T_{\mu\nu}=\sum_{a}\partial_{\mu}\phi^{a}\,\partial_{\nu}\phi^{a}-g_{\mu\nu}\Big{(}$
$\displaystyle\frac{1}{2}\sum_{a}\partial_{\alpha}\phi^{a}\,\partial_{\beta}\phi^{a}\,g^{\alpha\beta}$
(2) $\displaystyle-V(\phi^{a})\Big{)}=0.$

There is an analytic solution to Eq. (2) for $g_{\mu\nu}$ in this toy model:

$g_{\mu\nu}[\phi^{a}(x)]=\left(d/2-1\right)\frac{\sum_{a}\partial_{\mu}\phi^{a}\,\partial_{\nu}\phi^{a}}{V(\phi^{a})}.$
(3)

Eq. (3) defines the composite metric operator. Substituting the composite
metric into the action Eq. (1) gives the background-independent action that
serves to define the theory:

$S=\int
d^{d}x\,\left(\frac{d/2-1}{V(\phi^{a})}\right)^{\frac{d}{2}-1}\sqrt{\rm
det\left(\sum_{a}\partial_{\mu}\phi^{a}\,\partial_{\nu}\phi^{a}\right)}.$ (4)

A fundamental spacetime metric does not appear in Eq. (4), which we take as
the starting point for the remainder of the analysis.

Expanding the composite operators in Eq. (4) about a presumed vacuum
expectation value, it is possible to compute scattering amplitudes by summing
over the leading Feynman diagrams in a $1/N$ expansion. Such analyses have
confirmed the emergence of a gravitational interaction in vacuum composite
spacetimes with and without cosmological constant [7, 9, 8, 10, 11, 12]. In
all of these analyses, either dimensional regularization or Pauli-Villars
fields were used as a proxy for the presumed physical regulator.The stochastic
composite gravity framework completes the composite gravity scenario by
providing a physical ultraviolet regulator responsible for the emergent
gravitational interaction.

##  III Stochastic Field Theory

The framework introduces a discreteness into the stochastic description of
quantum mechanics developed by Nelson [19] and extended to quantum field
theory by Guerra and Ruggiero [20, 21], an idea proposed as a path towards
quantum gravity in Refs. [13]. The spacetime coordinates remain continuous,
but the set of events at which the fields experience stochastic fluctuations
is taken to be discrete. The ontology of the theory is straightforward, and
includes only physical fields whose dynamics include discrete random
fluctuations. States are described by stochastic processes with generally
nonlocal kinematics (which describes entanglement), but satisfying local
dynamical equations of motion.

We first briefly review the stochastic description of free scalar quantum
fields $\phi^{a}$. In stochastic field theory, states correspond to stochastic
processes described by a Langevin-type equation of the form,

$d\phi^{a}(\mathbf{x})=b_{\mathbf{x}}[\\{\phi^{b}(\mathbf{x})\\},t]\,dt+\sqrt{2D}\,dW^{a}_{t}(\mathbf{x}),$
(5)

where the drift function $b_{\mathbf{x}}$ is a possibly nonlocal function of
the fields and $t$; the diffusion parameter $D$ is for now taken to be
constant (but will later be effectively scale dependent); and $W_{t}$ is the
continuum Wiener process with expectation values,

$\displaystyle\mathbb{E}[dW^{a}_{t}(\mathbf{x})]$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\mathbb{E}[dW^{a}_{t}({\mathbf{x}})\,dW^{b}_{t}({\mathbf{x}^{\prime}})]$
$\displaystyle=$
$\displaystyle\delta^{3}({\mathbf{x}}-{\mathbf{x}^{\prime}})\delta^{ab}\,dt.$
(6)

We use the Ito convention, so that $dW_{t}$ at time $t$ represents a
contribution to the change in the field during a step $dt$ to the future of
$t$. We note that a primitive notion of causality is built into the language
of stochastic processes in this way.

It is convenient in the Lorentz-invariant theory to decompose the fields in a
complete orthonormal basis of modes $u_{\mathbf{k}}(\mathbf{x})$ with
$\phi^{a}(x)=\sum_{\mathbf{k}}\phi_{\mathbf{k}}(t)\,u_{\mathbf{k}}(\mathbf{x})$.
The modes satisfy the stochastic differential equation,

$d\phi^{a}_{\mathbf{k}}=b^{a}_{\mathbf{k}}[\\{\phi^{b}\\},t]\,dt+\sqrt{2D}\,dW^{a}_{\mathbf{k}},$
(7)

where $\mathbb{E}[dW_{\mathbf{k}}^{a}]=0$ and at each time $t$

$\mathbb{E}[dW^{a}_{\mathbf{k}}\,dW^{b}_{\mathbf{k}^{\prime}}]=\delta_{\mathbf{k}\mathbf{k}^{\prime}}\delta^{ab}\,dt.$
(8)

Following Nelson we also consider the backwards process:

$d\phi^{a}_{\mathbf{k}}=b^{a}_{*\mathbf{k}}[\\{\phi^{b}\\},t]\,dt+\sqrt{2D}\,dW^{a}_{*\mathbf{k}},$
(9)

with $dt<0$ representing evolution backwards in time, and
$dW^{a}_{*\mathbf{k}}$ distributed as in Eq. (8) but representing changes to
the past of $t$. Note that in general $b_{*\mathbf{k}}^{a}\neq
b_{\mathbf{k}}^{a}$.

Due to the nondifferentiable stochastic evolution in time, dynamics is most
easily expressed in terms of smooth expectation values. The stochastic forward
and backward derivatives at time $t$, denoted $D_{+}$ and $D_{-}$,
respectively, are defined in terms of expectation values over filtrations with
$\phi(\mathbf{x};t)$ fixed at time $t$:

$\displaystyle D_{+}f(\phi):=\lim_{\Delta t\rightarrow
0}\mathbb{E}\left[\frac{f(\phi(t+\Delta t))-f(\phi(t))}{\Delta t}\right]$ (10)
$\displaystyle D_{-}f(\phi):=\lim_{\Delta t\rightarrow
0}\mathbb{E}\left[\frac{f(\phi(t))-f(\phi(t-\Delta t))}{\Delta t}\right]$ (11)

Unlike for classical functions, in general $D_{+}f\neq D_{-}f$. The modes
satisfy a generalization of the free-field equations of motion suitable for
stochastic dynamics [20, 21]

$\frac{1}{2}\left(D_{+}D_{-}\phi^{a}_{\mathbf{k}}+D_{-}D_{+}\phi^{a}_{\mathbf{k}}\right)+\omega_{\mathbf{k}}^{2}\,\phi^{a}_{\mathbf{k}}=0,$
(12)

with $\omega_{\mathbf{k}}^{2}=\left(\mathbf{k}^{2}+m^{2}/\hbar^{2}\right)$.
The combination of $D_{+}$ and $D_{-}$ follows from time reversibility of the
resulting dynamics [19]. Evaluating the stochastic derivatives by expanding
the field about time $t$, one can write explicit expressions for the terms in
Eq. (12):

$\displaystyle D_{+}$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
t}+\sum_{a,\mathbf{k}}\,\left\\{b^{a}_{\mathbf{k}}\frac{\partial}{\partial\phi^{a}_{\mathbf{k}}}+D\,\frac{\partial^{2}}{\partial\phi_{\mathbf{k}}^{a\,2}}\right\\},$
$\displaystyle D_{-}$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
t}+\sum_{a,\mathbf{k}}\,\left\\{b^{a}_{*\mathbf{k}}\frac{\partial}{\partial\phi^{a}_{\mathbf{k}}}-D\,\frac{\partial^{2}}{\partial\phi_{\mathbf{k}}^{a\,2}}\right\\}.$
(13)

Unlike for classical functions, for functions of stochastic variables the
forward and backward derivatives are generally unequal. We note that

$D_{+}\phi^{a}_{\mathbf{k}}=b^{a}_{\mathbf{k}},\ \
D_{-}\phi^{a}_{\mathbf{k}}:=b^{a}_{*\mathbf{k}}.$ (14)

The convective velocity $v_{\mathbf{k}}^{a}$ and stochastic velocity
$u_{\mathbf{k}}^{a}$ are defined as,

$v_{\mathbf{k}}^{a}:=\frac{b_{\mathbf{k}}^{a}+b_{*\mathbf{k}}^{a}}{2},\ \ \
u_{\mathbf{k}}^{a}:=\frac{b_{\mathbf{k}}^{a}-b_{*\mathbf{k}}^{a}}{2}.$ (15)

By an H-theorem for stochastic processes [22], initial ensembles generically
evolve towards a common probability distribution, the quantum equilibrium
distribution satisfying the functional Schrödinger equation and the Born rule,
with the diffusion parameter now identified with $D=\hbar/2$.

Conversely, given a solution to the functional Schrödinger equation
$\psi[\phi(\mathbf{x}),t]=\sqrt{\rho}e^{iS}$, the stochastic process whose
probability density functional agrees with the Born rule $\rho=|\psi|^{2}$ for
that state has convective and stochastic velocities given by [19],

$v^{a}=2D\frac{\delta S}{\delta\phi^{a}},\ \ \
u^{a}=D\frac{\delta\ln\rho}{\delta\phi^{a}}.$ (16)

With knowledge of the convective and stochastic velocities, the forward drift
coefficient is then determined by $b^{a}=v^{a}+u^{a}$. The stochastic process
corresponding to the ground state of the free field is then found to be an
Ornstein-Uhlenbeck type process [20],

$d\phi^{a}(\mathbf{x})=-\sqrt{-\nabla^{2}+m^{2}/\hbar^{2}}\phi^{a}(\mathbf{x})\,dt+\sqrt{2D}\,dW^{a}(\mathbf{x}),$
(17)

or in terms of the mode expansion,

$d\phi^{a}_{\mathbf{k}}=-\sqrt{\mathbf{k}^{2}+m^{2}/\hbar^{2}}\phi^{a}_{\mathbf{k}}\,dt+\sqrt{2D}\,dW_{\mathbf{k}}^{a}.$
(18)

From Eqs. (15) and (16), stationary states in equilibrium, including the
vacuum, have $b_{*\mathbf{k}}^{a}=-b_{\mathbf{k}}^{a}$. From Eq. (18) we have
$b_{\mathbf{k}}^{a}=-\sqrt{\mathbf{k}^{2}+m^{2}/\hbar^{2}}\phi_{\mathbf{k}}^{a}$
in the vacuum state.

##  IV Stochastic Composite Gravity

Taking $V(\phi^{a})=V_{0}+\sum_{a}m^{2}/2\,:\phi^{a}\phi^{a}:$ and $d=4$,
where the normal-ordering indicates that the expectation value of the mass
term is absorbed in the definition of $V_{0}$, in the stochastic description
the action Eq. (4) is replaced by,

$S=\int
d^{4}x\,\mathbb{E}\left[\frac{\sqrt{\det(V_{0}G_{\mu\nu})}}{V(\phi^{a})}\right],$
(19)

where,

$\displaystyle G_{00}$
$\displaystyle=\frac{1}{2V_{0}}\sum_{a}\left(D_{+}\phi^{a}\,D_{+}\phi^{a}+D_{-}\phi^{a}\,D_{-}\phi^{a}\right)$
(20) $\displaystyle G_{i0}$
$\displaystyle=G_{0i}=\frac{1}{2V_{0}}\sum_{a}\left(D_{+}\phi^{a}+D_{-}\phi^{a}\right)\partial_{i}\phi^{a}$
$\displaystyle G_{ij}$
$\displaystyle=\frac{1}{V_{0}}\sum_{a}\partial_{i}\phi^{a}\,\partial_{j}\phi^{a}.$

With the approximation $V_{0}\gg m^{2}:\phi^{2}:$, at lowest order in
$1/V_{0}$ we have $g_{\mu\nu}\approx G_{\mu\nu}$. The equations of motion
follow from stationarization of the action over stochastic processes,
following the stochastic calculus of variations developed by Yasue [23]:

$D_{+}\frac{\delta S}{\delta(D_{-}\phi^{a})}+D_{-}\frac{\delta
S}{\delta(D_{+}\phi^{a})}+\partial_{i}\frac{\delta
S}{\delta(\partial_{i}\phi^{a})}-\frac{\delta S}{\delta\phi^{a}}=0.$ (21)

A version of Noether’s theorem applied to stochastic mechanics [23] allows us
to identify a conserved energy and momentum. Defining ${\cal L}$ through
$S=\int d^{4}x{\cal L}$, the energy-momentum tensor can be identified as:

$\displaystyle T_{0}^{\,0}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{a}\left(D_{+}\phi^{a}\frac{\partial{\cal
L}}{\partial(D_{+}\phi^{a})}+D_{-}\phi^{a}\frac{\partial{\cal
L}}{\partial(D_{-}\phi^{a})}\right)-{\cal L}\right]$ $\displaystyle
T_{0}^{\,i}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{a}\left(D_{+}\phi^{a}\frac{\partial{\cal
L}}{\partial(\partial_{i}\phi^{a})}+D_{-}\phi^{a}\frac{\partial{\cal
L}}{\partial(\partial_{i}\phi^{a})}\right)\right]$ $\displaystyle T_{i}^{\,0}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{a}\left(\partial_{i}\phi^{a}\frac{\partial{\cal
L}}{\partial(D_{+}\phi^{a})}+\partial_{i}\phi^{a}\frac{\partial{\cal
L}}{\partial(D_{-}\phi^{a})}\right)\right]$ $\displaystyle T_{i}^{\,j}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{a}\left(\partial_{i}\phi^{a}\frac{\partial{\cal
L}}{\partial(\partial_{j}\phi^{a})}\right)-\delta_{i}^{j}\,{\cal L}\right].$
(22)

It is straightforward to demonstrate that independent of the equations of
motion, $T_{\mu}^{\,\nu}=0$ in the model under consideration, confirming a
familiar consequence of diffeomorphism invariance. Solving the equations of
motion Eq. (21) is then analogous to solving the Wheeler-DeWitt equation of
canonical quantum gravity [24].

By a coordinate reparametrization we choose to arrange stochastic events via a
Poisson distribution per unit Cartesian volume, which then defines a local
reference frame. The Poisson distribution defines the regularization length
$\tau$. Poisson sprinklings (in infinite volume) respect Lorentz invariance
but not Euclidean invariance, because with a Euclidean metric the displacement
of nearest neighbors determines a preferred direction [25, 26].

Figure 1: The density of stochastic events in a Poisson sprinkling is Lorentz
invariant.

In terms of the microscopic description, Eq. (7) does not give a precise
description of the field evolution, but instead the Brownian contribution is
to be replaced by a discrete set of random $\delta$-function fluctuations at
the Poisson sprinkling of events. A dynamical consequence of this is that the
field in the equilibrium vacuum state has support localized around the
discrete set of stochastic events, resulting in an effective spacetime
discreteness. Eq. (17) then describes the evolution of the field after
averaging over a small three-volume as it traces out a 4-volume that includes
a large number of stochastic events. The density of stochastic events per unit
four-volume is Lorentz invariant, as illustrated in Fig. 1, so Eq. (17) takes
the same form in a Lorentz-transformed frame, with $\mathbf{x}$ and $t$
transformed appropriately.

In a mean field approximation in the continuous stochastic limit, we assume a
vacuum described approximately by the Lorentz-invariant vacuum process Eq.
(18), and we tune the constant $V_{0}$ in the potential so that
$\mathbb{E}[G_{\mu\nu}]$ is properly normalized.

To this end, we calculate $\mathbb{E}\left[G_{\mu\nu}\right]$ for the process
Eq. (18) in equilibrium. By Ito’s lemma, we have,

$\displaystyle d\mathbb{E}[(\phi_{\mathbf{k}}^{a})^{2}]$ $\displaystyle=$
$\displaystyle
2\,\mathbb{E}[\phi_{\mathbf{k}}^{a}\,d\phi_{\mathbf{k}}^{a}]+2D\,dt$ (23)
$\displaystyle=$
$\displaystyle-2\omega_{\mathbf{k}}\mathbb{E}[(\phi_{\mathbf{k}}^{a})^{2}]\,dt+2D\,dt,$

which can be integrated to give,

$\mathbb{E}[(\phi_{\mathbf{k}}^{a})^{2}]=\frac{D}{\omega_{\mathbf{k}}}+Ce^{-2\omega_{\mathbf{k}}t}.$
(24)

The constant $C$ depends on the initial field configuration, but independent
of $C$ the expectation value approaches a common equilibrium value. Similarly,
we find that in the infinite-volume, continuous process limit, the composite
metric approaches an equilibrium with

$\mathbb{E}\left[G_{\mu\nu}\right]=\frac{N}{V_{0}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{D\,k_{\mu}k_{\nu}}{\omega_{\mathbf{k}}}.$
(25)

Taking $D=\hbar/2$ we recognize this formally as the one-loop vacuum
expectation value of the composite operator $G_{\mu\nu}$ in the equivalent
Lorentz-invariant quantum field theory. The UV divergence is regularized by
discreteness of the stochastic process. As the discreteness scale is
approached, Eq. (7) no longer provides an accurate description of the
kinematics as the diffusive effects diminish. This can be modeled by replacing
the diffusion parameter $D$ with a scale-dependent $D(\omega_{\mathbf{k}})$.
If we take the typical discreteness scale in the Poisson sprinkling to be
$\tau\ll\hbar/m$, then a natural choice for $D(\omega_{\mathbf{k}}$) would be
$D(\omega_{\mathbf{k}})=e^{-\omega_{\mathbf{k}}\tau}$, though we should
average over timelike directions to capture the Lorentz invariance of the
description.

With this replacement of $D$ in Eq. (25), the spacetime discreteness acts as a
Euclidean-time point splitting regulator of the one-loop integral in the
effective quantum field theory. We recall that correlation functions of
products of fields in stochastic field theory have a direct physical
interpretation and are equivalent to Wick rotations of the Wightman functions
in the corresponding quantum field theory, as was noted by Guerra and Ruggiero
[20, 21]. Hence, we can interpret the above result as equivalent to the
Lorentzian point splitting of the composite operator $G_{\mu\nu}$ in the
quantum field theory. Fig. 2a illustrates a simulation of $G_{00}$ with a far-
from-equilibrium initial field configuration and demonstrates the evolution
towards a spatially uniform equilibrium configuration.

We see that locally there are large fluctuations of the composite metric even
after equilibrium is reached, but $G_{00}$ appears uniform when averaged over
regions of several volumes associated with the discreteness scale.

Figure 2: a) Evolution of the composite operator $G_{00}$ from a far-from-
equilibrium initial field configuration over $10^{3}$ time steps in 2+1
dimensions including the lowest $20\times 20$ even-parity modes. b) $G_{00}$
in the final configuration from (a), but averaged around each point over
$5\times 5$ spatial volumes in units of the time step. $V_{0}$ can be tuned so
that the uniform $G_{00}(\mathbf{x})$ in equilibrium is normalized to 1 self-
consistently with the presumed Minkowski space vacuum.

To demonstrate the emergence of a gravitational interaction about the mean-
field vacuum we would reproduce the analysis of Refs. [7, 12] with dimensional
regularization replaced by the stochastic description. A detailed analysis
will appear elsewhere. However, we briefly comment on the scales associated
with the emergent gravitational interaction. Refs. [7, 12] identify the
gravitational interaction by evaluation of the 2$\rightarrow$2 scattering
amplitude at leading order in $1/N$. The relevant chains of loop integrals
that arise are summed by evaluating a kernel proportional to $\langle{\cal
T}_{\mu\nu}(x){\cal T}_{\alpha\beta}(y)\rangle$, where ${\cal T}_{\mu\nu}$ is
the energy-momentum tensor for the free field in the vacuum spacetime. With
the stochastic regulator, the leading divergences of the integral are ${\cal
O}(N/\tau^{4})$ and ${\cal O}(Nq^{2}/\tau^{2})$. These divergences lead to a
Planck mass $M_{\rm Pl}\sim\sqrt{N}/\tau$.

The effective Planck mass is hierarchically larger than the regulator scale,
so the effective quantum field theory description should give way to smooth
field evolution before dangerous Planck-scale effects are manifested. This
also suggests the suppression of higher-dimension operators in the effective
theory at all scales.

##  V Conclusions

We have presented a first analysis of an approach to quantum gravity in which
discreteness of random fluctuations in a stochastic representation of quantum
field theory provides a physical UV regulator. We found that, with a tuning of
the constant in the potential, there is a self-consistent vacuum in which the
expectation value of the composite metric operator fluctuates about a flat
metric. We demonstrated that the conserved energy and momentum defined through
an analogy to Noether’s theorem for stochastic mechanics vanish identically,
as expected in a metric-independent theory. We argued that fluctuations about
the vacuum should give rise to an effective gravitational interaction as in
prior analyses of composite gravity models that used standard nonphysical UV
regulators as a proxy for a physical regulator like that provided by the
stochastic field theory.

The absence of gravitation at ultra-short distances might be relevant for
understanding the apparent low entropy of the early universe [27].
Furthermore, the possibility of a far-from-equilibrium initial field
configuration leads to the possibility of imprints in early cosmology and on
the horizon scale, for example by analogy with the prediction of a decreased
power in the low-$\ell$ moments of the CMB power spectrum due to
nonequilibrium effects in Bohmian field theory as demonstrated by Valentini
[28]. Transitions between atomic states might also lead to temporary far-from-
equilibrium configurations that would suppress further transitions when probed
with ultrafast sequences of laser pulses tuned to atomic transition
frequencies.

As a nascent framework for quantum gravity, it will be important to revisit
the puzzles of semiclassical black hole physics, spacetime singularities and
cosmology. As gravitation arises only at distances longer than the
discreteness scale in stochastic composite gravity, we do not necessarily
expect or require that a microscopic counting of degrees of freedom would
reproduce the Bekensein-Hawking black hole entropy. Cosmological spacetimes
would need to be described by nonperturbative solutions to the stochastic
field equations, but may be amenable to a self-consistent mean-field analysis
as was presented here for a flat-space vacuum. It may prove interesting to
consider how the spectral dimension of the theory varies with scale [29]. It
will also be necessary to extend the stochastic framework to include fermions
and gauge fields. To this end, we might look towards the stochastic
quantization framework of Parisi and Wu, which includes fermions and gauge
fields [30, 31], for guidance.

###### Acknowledgements.

This work was supported in part by the NSF under Grant PHY-1819575 and by a
Plumeri Award for Faculty Excellence at William & Mary.

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