#  Proto-CLIP: Vision-Language Prototypical Network for Few-Shot Learning

Jishnu Jaykumar P1 Kamalesh Palanisamy1 Yu-Wei Chao2 Xinya Du1 Yu Xiang1
1The University of Texas at Dallas 2NVIDIA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose a novel framework for few-shot learning by leveraging large-scale
vision-language models such as CLIP [1]. Motivated by unimodal prototypical
networks for few-shot learning, we introduce Proto-CLIP that utilizes image
prototypes and text prototypes for few-shot learning. Specifically, Proto-CLIP
adapts the image encoder and text encoder in CLIP in a joint fashion using
few-shot examples. The two encoders are used to compute prototypes of image
classes for classification. During adaptation, we propose aligning the image
prototypes and the text prototypes of the corresponding classes. Such
alignment is beneficial for few-shot classification due to the reinforced
contributions from both types of prototypes. We demonstrate the effectiveness
of our method by conducting experiments on benchmark datasets for few-shot
learning, as well as in the real world for robot perception111Project page is
available at https://irvlutd.github.io/Proto-CLIP.

> Keywords: Robot Perception, Object Recognition, Few-Shot Learning,
> Contrastive-Learning, Vision-Language, Multimodal

##  1 Introduction

Building autonomous robots that can help people perform various tasks is the
dream of every roboticist. Nowadays, most robots are working in factories and
warehouses by performing pre-programmed repetitive tasks such as assembling
and delivering. In the future, we believe that there will be intelligent
robots that can perform tasks in human environments autonomously. For example,
people can instruct a robot by saying “bring me a bottle of water” or “wash
the mug on the table”, then the robot will execute the instructions
accordingly. In these scenarios, robots need to recognize objects from sensory
data in order to understand the required tasks. In this work, we develop a
novel few-shot learning method that can enable robots to recognize novel
objects from just a few example images per object.

We believe that few-shot learning [2] is a promising paradigm to enable robots
to recognize a large number of objects. The appeal lies in the ease of data
collection—just a few example images is sufficient for teaching a robot a
novel object. On the contrary, object model-based approaches build 3D models
of objects and then use these 3D models [3] for object recognition. Object
category-based approaches focus on recognizing category labels of objects such
as 80 categories in the MSCOCO dataset [4]. The limitation of model-based
object recognition is the difficulty of obtaining a large number of 3D models
for many objects in the real world. Current 3D scanning techniques cannot deal
well with metal objects or transparent objects. For category-based object
recognition, it is difficult to obtain a large number of images for each
category in robotic settings. Large-scale datasets for object categories such
as ImageNet [5] and Visual Genome [6] are collected from the Internet. These
Internet images are not very suitable for learning object representations for
robot manipulation due to domain differences. Due to the limitations of model-
based and category-based object recognition, if a robot can learn to recognize
a new object from a few images of the object, it is likely to scale up the
number of objects that the robot can recognize in the real world.

Figure 1: Our Proto-CLIP model learns a joint embedding space of images and
text, where image prototypes and text prototypes are learned and aligned using
support sets for few-shot classification.

The main challenge in few-shot learning is how to achieve generalization with
very limited training examples. Learning good visual representations is the
key to achieve good performance in few-shot learning [7]. Although the
Internet images are quite different from robot manipulation settings, they can
be used to learn powerful visual representations. Recently, the CLIP
(Contrastive Language–Image Pre-training) model [1] trained with a large
number of image-text pairs from the Internet achieves promising _zero-shot_
image recognition performance. Using the visual and language representations
from CLIP, several few-shot learning approaches [8, 9, 10] are proposed to
improve the zero-shot CLIP model. [9, 10] adapt the CLIP image encoder to
learn better feature representations, while [8] learns prompts for the CLIP
model. On the other hand, few-shot learning approaches are studied in the
meta-learning framework [11]. These approaches are aimed at generalizing to
novel classes after training. A notable method is Prototypical Network [12]
and its variants [13, 14], which demonstrate effective performance for few-
shot learning. However, these methods do not leverage the powerful feature
representation of CLIP.

These observations motivate us to leverage CLIP in prototypical networks for
few-shot learning. We notice that existing methods for adapting CLIP models in
few-shot learning adapt the image encoder [9, 10] or the text encoder [8] in
CLIP. We argue that if we can use both the image encoder and the text encoder
for classification and jointly adapt them using few-shot training images, we
can improve the few-shot classification performance. To achieve this goal, we
propose Proto-CLIP, a new model motivated by the traditional unimodal
Prototypical Networks [12]. Proto-CLIP utilizes image prototypes and text
prototypes computed from adapted CLIP encoders for classification. In
addition, we propose to align the image prototype and the text prototype of
the same class during adaptation. In this way, both the image encoder and the
text encoder can contribute to the classification while achieving agreement
between their predictions. Fig. 1 illustrates the concept of learning the
joint embedding space of images and text from Proto-CLIP.

To verify the effectiveness of Proto-CLIP, we have conducted experiments on
commonly used benchmarks for few-shot image classification, as well as the
FewSOL dataset introduced for few-shot object learning in robotic environments
[15]. In addition, we have built a robotic system that integrates Automatic
Speech Recognition (ASR), few-shot object recognition using Proto-CLIP and
robotic grasping to demonstrate the robotic application of Proto-CLIP.

##  2 Related Work

In the context of image recognition, few-shot learning indicates using a few
images per image category. The problem is usually formulated as “$N$-way,
$K$-shot”, i.e., $N$ classes with $K$ images per class. In the traditional
image classification setup, these $NK$ images are used as training images.
Once a model is trained, it can be used to test images among $N$ classes.
Recent CLIP-based few-shot learning methods fall into this setting.

Method Use Support Sets Adapt Image Embedding Adapt Text Embedding Align Image
and Text Zero-shot CLIP [1] ✗ ✗ ✗ ✓ Linear-probe CLIP [1] ✓ ✓ ✗ ✗ CoOp [8] ✓ ✗
✓ ✗ CLIP-Adapter [9] ✓ ✓ ✓ ✗ Tip-Adapter [10] ✓ ✓ ✗ ✗ Sus-X [16] ✓ ✓ ✗ ✗
Proto-CLIP (Ours) ✓ ✓ ✓ ✓

Table 1:  Comparison between our proposed method with existing CLIP-based
methods for few-shot learning. “Use Support Sets” indicates if a method uses
support training sets for fine-tuning. “Adapt Image/Text Embedding” indicates
if a method adapts the image/text embeddings in CLIP. “Align Image and Text”
indicates if a method specifically aligns images and text in the feature
space.

CLIP-based Few-Shot Learning. The CLIP [1] model applies contrastive learning
to image-text pairs from the Internet. It consists of an image encoder and a
text encoder for the extraction of features from images and text,
respectively. Its training objective is to maximize the similarity between the
corresponding image and text in a pair in a high-dimensional joint feature
space. After training, CLIP can be used for zero-shot image classification by
comparing image features with text embeddings of novel class names. This model
is denoted as zero-shot CLIP. When a few training images are available for
each class, several approaches are proposed to improve zero-shot CLIP. The
linear-probe CLIP model [1] trains a logistic regression classifier using CLIP
image features. CoOp [8] proposes to use learnable vectors as a prompt for the
CLIP text encoder for few-shot learning. CLIP-Adapter [9] learns two layers of
linear transformations on top of the image encoder and the text encoder with
residual connections, respectively, to adapt CLIP features for few-shot
learning. Tip-Adapter [10] builds a key-value cache model, where keys are CLIP
image features and values are one-hot vectors of the class labels. Given a
query image, its image feature is compared with the cache keys to combine the
value labels for classification. Tip-Adapter can also fine-tune the keys by
treating them as learnable parameters, which further improves the few-shot
classification accuracy. Sus-X [16] leverages the power of Stable Diffusion
[17] to create support sets and aims to address the issue of uncalibrated
intra-modal embedding distances in TIP-Adapter [10] by utilizing inter-modal
distances as a connecting mechanism.

Table 1 compares our proposed method with existing CLIP-model-based few-shot
learning methods. By using the image prototypes and text prototypes for
classification, our method can adapt both the image embeddings and text
embeddings from CLIP. In addition, the model aligns the image prototypes and
the text prototypes, which serves as a regularization term in adapting the
feature embeddings. We empirically verify our model by conducting experiments
on benchmark datasets for few-shot learning.

Meta-learning-based Few-Shot Learning. In parallel with these efforts to adapt
CLIP for few-shot learning, meta-learning-based approaches are also proposed
for few-shot learning. While previous CLIP-based models are tested on the same
classes in training, the focus here is to learn a model on a set of training
classes $\mathcal{C}_{train}$ that can generalize to novel classes
$\mathcal{C}_{test}$ in testing. Each class contains a support set and a query
set. During training, the class labels for both sets are available. During
testing, only the class labels of the support set are available, and the goal
is to estimate the labels of the query set. Meta-learning-based approaches
train a meta-learner with the training classes $\mathcal{C}_{train}$ that can
be adapted to the novel classes $\mathcal{C}_{test}$ using their support sets.
Non-episodic approaches use all the data in $\mathcal{C}_{train}$ for training
such as $k$-NN and its ‘Finetuned’ variants [18, 19, 20, 7]. Episodic
approaches construct episodes, i.e., a subset of the training classes, to
train the meta-learner. Representative episodic approaches include
Prototypical Networks [12], Matching Networks [21], Relation Networks [22],
Model Agnostic Meta-Learning (MAML) [11], Proto-MAML [13] and
CrossTransformers [14]. The Meta-Dataset [13] was introduced to benchmark few-
shot learning methods in this setting. In this work, we consider training and
testing in the same classes following previous CLIP-based few-shot learning
methods [8, 9, 10].

##  3 Method

We consider the $N$-way $K$-shot classification problem. In few-shot settings,
$K\ll N$. The image set with class labels is considered as the _support set_ :
$\mathcal{S}=\\{\mathbf{x}_{i}^{s},y_{i}^{s}\\}_{i=1}^{M}$, where
$\mathbf{x}_{i}^{s}$ denotes a support image, $y_{i}^{s}\in\\{1,2,\ldots,N\\}$
denotes the class label of the support image, and $M$ is the size of the
support set. In $N$-way $K$-shot settings, $M=NK$. The goal of few-shot
classification is to classify the _query set_
$\mathcal{Q}=\\{\mathbf{x}_{j}^{q}\\}_{j=1}^{L}$, i.e., $L$ test images
without class labels. Specifically, we want to estimate the conditional
probability $P(y=k|\mathbf{x}^{q},\mathcal{S})$ that models the probability
distribution of the class label $y$ given a query image $\mathbf{x}^{q}$ and
the support set $\mathcal{S}$.

Figure 2: Overview of our proposed Proto-CLIP model. The CLIP image encoder
and text encoder are frozen during training. The image memory, the text memory
and the adapter network are learned. Given a class name, $\tau_{i}$ returns
the $i^{th}$ out of $\tilde{K}$ predefined text prompts.

Our Proto-CLIP model (Fig. 2). We propose to leverage both the image encoder
and the text encoder in the CLIP model [1] to estimate the conditional
probability of class label as

$P(y=k|\mathbf{x}^{q},\mathcal{S})=\alpha\underbrace{P(y=k|\mathbf{x}^{q},\mathcal{S}_{x})}_{\text{image
probability}}+(1-\alpha)\underbrace{P(y=k|\mathbf{x}^{q},\mathcal{S}_{y})}_{\text{text
probability}},$ (1)

where $\mathcal{S}_{x}=\\{\mathbf{x}_{i}^{s}\\}_{i=1}^{M}$ and
$\mathcal{S}_{y}=\\{y_{i}^{s}\\}_{i=1}^{M}$ denote the image set and the label
set of the support set $\mathcal{S}$, respectively, and $\alpha\in[0,1]$ is a
hyper-parameter to combine the two probabilities. To model the probability
distributions conditioned on $\mathcal{S}_{x}$ or $\mathcal{S}_{y}$, we
leverage the prototypical networks [12]:

$\displaystyle
P(y=k|\mathbf{x}^{q},\mathcal{S}_{x})=\frac{\exp(-\beta\|g_{\mathbf{w}_{1}}(\mathbf{x}^{q})-\mathbf{c}_{k}^{x}\|_{2}^{2})}{\sum_{k^{\prime}=1}^{N}\exp(-\beta\|g_{\mathbf{w}_{1}}(\mathbf{x}^{q})-\mathbf{c}_{k^{\prime}}^{x}\|_{2}^{2})},$
(2) $\displaystyle
P(y=k|\mathbf{x}^{q},\mathcal{S}_{y})=\frac{\exp(-\beta\|g_{\mathbf{w}_{1}}(\mathbf{x}^{q})-\mathbf{c}_{k}^{y}\|_{2}^{2})}{\sum_{k^{\prime}=1}^{N}\exp(-\beta\|g_{\mathbf{w}_{1}}(\mathbf{x}^{q})-\mathbf{c}_{k^{\prime}}^{y}\|_{2}^{2})},$
(3)

where $g_{\mathbf{w}_{1}}(\cdot)$ denotes the CLIP image encoder plus an
adapter network with learnable parameters $\mathbf{w}_{1}$ used to compute the
feature embeddings of query images. The CLIP image encoder is pretrained and
then frozen. $\mathbf{c}_{k}^{x}$ and $\mathbf{c}_{k}^{y}$ are the
“prototypes” for class $k$ computed using images and text, respectively.
$\beta\in\mathbb{R}^{+}$ is a hyperparameter to sharpen the probability
distributions. We have the prototypes as

$\mathbf{c}_{k}^{x}=\frac{1}{M_{k}}\sum_{y_{i}^{s}=k}\phi_{\text{Image}}(\mathbf{x}_{i}^{s}),\;\;\mathbf{c}_{k}^{y}=\frac{1}{\tilde{M_{k}}}\sum_{j=1}^{\tilde{M_{k}}}\phi_{\text{Text}}(\text{Prompt}_{j}(y_{i}^{s}=k)),$
(4)

where $M_{k}$ is the number of examples with label $k$, and $\tilde{M_{k}}$ is
the number of prompts for label $k$. To compute text embeddings, we can either
directly input the class names such as “mug” and “plate” into the text
encoder, or convert the class names to phases such as “a photo of mug” and “a
photo of plate” and then input the phrases into the text encoder. These
phrases are known as _prompts_ of the vision-language models. We can use
multiple prompts for each class label.
$\phi_{\text{Image}}(\mathbf{x}_{i}^{s})$ and
$\phi_{\text{Text}}(\text{Prompt}_{j}(y_{i}^{s}=k))$ denote the image
embedding and the text embedding of the image-label pair
$(\mathbf{x}_{i}^{s},y_{i}^{s})$ computed using the CLIP image encoder and the
text encoder, respectively. These embeddings with dimension $C$ of the support
set form the image memory and the text memory, as shown in Fig. 2. They are
learnable embedding vectors initialized by the computed embeddings using the
CLIP image encoder and text encoder. We use $\mathbf{c}_{k}^{x}$ and
$\mathbf{c}_{k}^{y}$ to denote the mean of the embeddings of the images and
the prompts for class $k$, respectively. Since the image embeddings and the
text embeddings are of the same dimension, we can compute the distance between
the text prototype $\mathbf{c}_{k}^{y}$ and the image embedding
$g_{\mathbf{w}_{1}}(\mathbf{x}^{q})$ in Eq. (3). As we can see, our model
leverages prototypical networks with image encoder and text encoder from CLIP.
We name it “Proto-CLIP”.

Figure 3: Two designs of the adapters. (a) A Multi-layer perceptron-based
adapter as in [9]. (b) A convolution-based adapter we introduce. The feature
dimension is for CLIP ResNet50 backbone.

Learning the memories and the adapter. During training, we can construct a
support set $\mathcal{S}=\\{\mathbf{x}_{i}^{s},y_{i}^{s}\\}_{i=1}^{M}$ and a
query set with ground truth labels
$\mathcal{Q}=\\{\mathbf{x}_{j}^{q},y_{j}^{q}\\}_{j=1}^{L}$. Then we can use
$\mathcal{S}$ and $\mathcal{Q}$ to learn the weights in Proto-CLIP. First, the
support set is used to initialize the image memory $\mathbf{W}_{\text{image}}$
and the text memory $\mathbf{W}_{\text{text}}$. Second, the weights in the
adapter network applied to the query images $g_{\mathbf{w}_{1}}(\cdot)$ need
to be learned. Fig. 3 shows two designs of the adapter network, i.e., an MLP-
based adapter as in [9] and a convolution-based adapter we introduce. The
convolution-based adapter has fewer weights to learn compared to the MLP-based
one. We found that the two adapters have their own advantages on different
datasets in our experiments. Finally, motivated by the CLIP-Adapter [9], we do
not fine-tune the weights in the image encoder and text encoder by freezing
these weights during training. In this way, we can reuse the weights of CLIP
trained on a large number of image-text pairs and adapt the image embeddings
and the text embeddings.

Loss Functions. The first loss function is the negative log-probability of the
true label for a query image:
$\mathcal{L}_{1}(\mathbf{W}_{\text{image}},\mathbf{W}_{\text{text}},\mathbf{w}_{1})=-\log
P(y^{q}=k|\mathbf{x}^{q},\mathcal{S})$, where
$P(y^{q}=k|\mathbf{x}^{q},\mathcal{S})$ is defined in Eq. (1). Minimizing
$\mathcal{L}_{1}$ learns the weights to classify the query images correctly.
Second, we propose aligning the image prototypes and the text prototypes in
training. Let
$\\{\mathbf{c}_{1}^{x},\mathbf{c}_{2}^{x},\ldots,\mathbf{c}_{N}^{x}\\}$ be the
image prototypes computed from the image embeddings for the $N$ classes and
$\\{\mathbf{c}_{1}^{y},\mathbf{c}_{2}^{y},\ldots,\mathbf{c}_{N}^{y}\\}$ be the
corresponding text prototypes. We would like to learn the model weights such
that $\mathbf{c}_{k}^{x}$ is close to $\mathbf{c}_{k}^{y}$ and far from other
prototypes in the embedding space. We utilize the InfoNCE loss for contrastive
learning [23]:

$\mathcal{L}_{2}^{k}(\mathbf{c}_{k}^{x},\\{\mathbf{c}_{k^{\prime}}^{y}\\}_{k^{\prime}=1}^{N})=-\log\frac{\exp(\mathbf{c}_{k}^{x}\cdot\mathbf{c}_{k}^{y})}{\sum_{k^{\prime}=1}^{N}\exp(\mathbf{c}_{k}^{x}\cdot\mathbf{c}_{k^{\prime}}^{y})},\mathcal{L}_{3}^{k}(\mathbf{c}_{k}^{y},\\{\mathbf{c}_{k^{\prime}}^{x}\\}_{k^{\prime}=1}^{N})=-\log\frac{\exp(\mathbf{c}_{k}^{y}\cdot\mathbf{c}_{k}^{x})}{\sum_{k^{\prime}=1}^{N}\exp(\mathbf{c}_{k}^{y}\cdot\mathbf{c}_{k^{\prime}}^{x})}$
(5)

for $k=1,\dots,N$, where $\cdot$ indicates dot-product. Here,
$\mathcal{L}_{2}^{k}(\mathbf{c}_{k}^{x},\\{\mathbf{c}_{k^{\prime}}^{y}\\}_{k^{\prime}=1}^{N})$
compares an image prototype $\mathbf{c}_{k}^{x}$ with the text prototypes
$\\{\mathbf{c}_{k^{\prime}}^{y}\\}_{k^{\prime}=1}^{N}$, while
$\mathcal{L}_{3}^{k}(\mathbf{c}_{k}^{y},\\{\mathbf{c}_{k^{\prime}}^{x}\\}_{k^{\prime}=1}^{N})$
compares a text prototype $\mathbf{c}_{k}^{y}$ with the image prototypes
$\\{\mathbf{c}_{k^{\prime}}^{x}\\}_{k^{\prime}=1}^{N}$. In this way, we can
align the image prototypes and the text prototypes for the $N$ classes. This
alignment can facilitate classification, since the class conditional
probabilities are computed using the image prototypes and the text prototypes
as in Eqs. (2) and (3). The total loss function for training is:

$\displaystyle\mathcal{L}=-\frac{1}{L}\sum_{j=1}^{L}\log
P(y_{j}^{q}=k|\mathbf{x}_{j}^{q},\mathcal{S})+\frac{1}{N}\sum_{k=1}^{N}\big{(}\mathcal{L}_{2}^{k}(\mathbf{c}_{k}^{x},\\{\mathbf{c}_{k^{\prime}}^{y}\\}_{k^{\prime}=1}^{N})+\mathcal{L}_{3}^{k}(\mathbf{c}_{k}^{y},\\{\mathbf{c}_{k^{\prime}}^{x}\\}_{k^{\prime}=1}^{N})\big{)}$
(6)

for a query set $\mathcal{Q}=\\{\mathbf{x}_{j}^{q},y_{j}^{q}\\}_{j=1}^{L}$.
Following previous CLIP-based few-shot learning methods [8, 9, 10], the
support set and the query set are the same during training in our experiments,
i.e., $\mathcal{S}=\mathcal{Q}$.

##  4 Experiments

Datasets and Evaluation Metric. Following previous CLIP-based few-shot
learning methods [8, 9, 10], we conduct experiments on the following datasets
for evaluation: ImageNet [5], StandfordCars [24], UCF101 [25], Caltech101
[26], Flowers102 [27], SUN397 [28], DTD [29], EuroSAT [30], FGVCAircraft [31],
OxfordPets [32], and Food101 [33]. In addition, we also include the FewSOL
dataset [15] recently introduced for few-shot object recognition in robotic
environments. In the $N$-way $K$-shot classification setting, $K$ images for
each class will be sampled from each dataset for training. A validation set of
each dataset is reserved for hyper-parameter tuning, and a test set is used
for evaluation. We report the classification accuracy of the test set as an
evaluation metric.

Choosing the Hyper-parameters: $\alpha$ and $\beta$. From the experiments, we
found that the two hyper-parameters $\alpha$ in Eq. (1) and $\beta$ in Eqs.
(2) and (3) play a critical role in classification accuracy. Therefore, for
each dataset, we conducted a grid search of the two parameters using the
validation set. Then we finalize their values for all the runs in our
experiments.

Proto-CLIP Variants. i) “Proto-CLIP”: we do not train the image memory and the
text memory and do not use any adapter in Proto-CLIP (Fig. 2), we directly run
inference using the pre-trained CLIP features. ii) “Proto-CLIP-$F$”: we train
the image memory and/or the text memory with the adapter. During training, for
all the query images, we precompute their CLIP image features and directly use
these stored features for training. This variant can be trained more quickly.
Therefore, we use it for our ablation studies. iii) “Proto-CLIP-$F$-$Q^{T}$”:
During training, for each query image, we apply random data augmentation
operations such as cropping and horizontal flip. Then we compute CLIP image
features for the transformed query images during training.

Figure 4:  Barnes-Hut t-SNE visualization [34] using the FewSOL dataset [15].
(a) Image and text prototypes from zero-shot CLIP, which are not aligned. (b)
Aligned image and text prototypes from Proto-CLIP-$F$.

###  4.1 Ablation Studies

Adapter Types and Learnable Text Memory. Since the 12 datasets have different
characteristics, we found that varying adapter types and whether to learn the
text memory or not affect performance. Table 2 summarizes the result of this
ablation study. The architectures of the MLP-based adapter and the
convolution-based adapter are illustrated in Fig. 3. “2xConv” indicates using
2 convolution layers as shown in Fig. 3, while “3xConv” uses 3 convolution
layers in the adapter where we add a $32@3\times 3\times 32$ convolution layer
in the middle. By checking the best accuracy for each dataset, we can see that
there is no consensus on which adapter and trainable text memory to use among
these datasets. Therefore, we select the best configuration on the adapter and
learnable text memory for each dataset in the following experiments. Learning
both image memory and text memory can obtain aligned image-text prototypes.
Fig. 4 visualizes the image-text prototypes in the FewSOL dataset [15] before
and after training.

Adapter | Train-Text-Memory | ImageNet | FGVC | Pets | Cars | EuroSAT | Caltech101 | SUN397 | DTD | Flowers | Food101 | UCF101 | FewSOL
---|---|---|---|---|---|---|---|---|---|---|---|---|---
MLP | ✗ | 61.06 | 35.31 | 85.61 | 72.19 | 83.47 | 92.58 | 68.54 | 63.89 | 95.01 | 74.05 | 76.16 | 28.65
MLP | ✓ | 61.06 | 37.56 | 85.72 | 73.61 | 83.53 | 92.13 | 69.71 | 63.89 | 96.06 | 74.05 | 76.16 | 32.87
2xConv | ✗ | 65.75 | 34.38 | 89.62 | 75.25 | 81.85 | 93.40 | 71.94 | 67.85 | 94.76 | 79.09 | 77.50 | 27.13
2xConv | ✓ | 58.60 | 35.82 | 89.21 | 74.34 | 81.78 | 93.02 | 69.79 | 67.32 | 95.82 | 78.06 | 76.37 | 27.13
3xConv | ✗ | 65.37 | 34.41 | 88.74 | 75.25 | 82.21 | 93.43 | 71.63 | 67.67 | 94.40 | 79.11 | 77.50 | 29.78
3xConv | ✓ | 59.63 | 36.15 | 87.93 | 72.68 | 81.57 | 92.74 | 68.64 | 68.56 | 95.78 | 78.61 | 77.03 | 35.22

Table 2: Ablation study of query adapters with $K=16$ and Proto-CLIP-$F$. In
all cases, the adapter and the visual memory keys are trained. In case of tie,
an underlined setup was used.

Loss functions. We have introduced three different loss functions in Sec. 3:
$\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3}$. We analyze the effects of
these loss functions in Table 3. We can see that i) the $\mathcal{L}_{1}$ loss
function is essential since it drives the classification of the query images;
ii) Overall, both $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$ loss functions for
prototype alignment contribute to the performance, which verifies our
motivation of aligning image and text prototypes for few-shot classification.

Loss | ImageNet | FGVC | Pets | Cars | EuroSAT | Caltech101 | SUN397 | DTD | Flowers | Food101 | UCF101 | FewSOL
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{L}_{1}$ | 62.67 | 20.34 | 73.21 | 73.77 | 78.98 | 92.25 | 68.34 | 66.49 | 96.14 | 77.39 | 76.66 | 34.57
$\mathcal{L}_{2}$ | 62.29 | 4.71 | 0.00 | 0.00 | 38.95 | 0.28 | 66.93 | 67.38 | 10.31 | 77.71 | 57.41 | 32.70
$\mathcal{L}_{3}$ | 62.27 | 4.14 | 0.00 | 0.00 | 38.09 | 0.24 | 64.86 | 67.38 | 10.27 | 77.69 | 57.55 | 20.22
$\mathcal{L}_{1}+\mathcal{L}_{2}$ | 65.39 | 36.24 | 88.58 | 75.39 | 82.78 | 93.71 | 71.65 | 68.09 | 96.06 | 78.69 | 77.29 | 33.48
$\mathcal{L}_{2}+\mathcal{L}_{3}$ | 62.33 | 3.87 | 0.00 | 0.00 | 36.86 | 0.24 | 64.84 | 68.32 | 8.20 | 77.35 | 57.52 | 19.61
$\mathcal{L}_{1}+\mathcal{L}_{3}$ | 65.43 | 36.84 | 88.58 | 75.51 | 82.84 | 93.35 | 71.44 | 68.32 | 96.14 | 78.80 | 77.53 | 33.43
$\mathcal{L}_{1}+\mathcal{L}_{2}+\mathcal{L}_{3}$ | 65.75 | 37.56 | 89.62 | 75.25 | 83.53 | 93.43 | 71.94 | 68.56 | 96.06 | 79.09 | 77.50 | 35.22

Table 3: Loss function ablation results with the ResNet50 backbone and shot
$K=16$.

Backbones. Table 4 shows the results of using different backbone networks on
the FewSOL dataset [15]. In general, better backbones can learn more powerful
feature representations and consequently improve the classification accuracy.
CLIP vision transformer backbones achieve better performance than CLIP ResNet
backbones.

Model | Adapter | TextM | Backbone
---|---|---|---
RN50 | RN101 | ViT-B/16 | ViT-B/32 | ViT-L/14
Zero-Shot-CLIP [1] | - | - | 25.91 | 32.96 | 40.70 | 41.87 | 54.57
Tip [10] | - | - | 29.74 | 37.43 | 47.00 | 41.48 | 56.78
Tip-F [10] | - | - | 32.52 | 41.43 | 50.17 | 45.48 | 60.17
Proto-CLIP-$F$ | MLP | ✗ | 33.48 | 39.04 | 47.96 | 41.91 | 58.65
Proto-CLIP-$F$ | MLP | ✓ | 34.83 | 40.74 | 47.43 | 42.13 | 58.91
Proto-CLIP-$F$ | 2xConv | ✗ | 35.04 | 41.04 | 50.83 | 46.52 | 63.74
Proto-CLIP-$F$ | 2xConv | ✓ | 35.04 | 42.52 | 49.26 | 43.43 | 61.61
Proto-CLIP-$F$ | 3xConv | ✗ | 34.13 | 42.83 | 51.91 | 46.87 | 62.35
Proto-CLIP-$F$ | 3xConv | ✓ | 35.22 | 44.09 | 50.39 | 46.57 | 60.39

Table 4: Backbone ablation study. Dataset=FewSOL-52 [15]. $K=16$. Model=Proto-
CLIP-$F$. “TextM” indicates whether to train text memory.

###  4.2 Comparison with Other Methods

Table 5 shows the performance of Proto-CLIP compared to the state-of-the-art
methods using CLIP for few-shot learning in the literature: Linear-Probe CLIP
[1], CoOp [8], CLIP-Adapter [9] and Tip-Adapter [8]. We follow these methods
and use ResNet50 backbone for this comparison. The fine-tuned variant of Tip-
Adapter “Tip-F” is the most competitive method compared to ours. The
performance of Proto-CLIP on very few shots, i.e., 1 shot and 2 shots is
inferior compared to Tip-F. When the number of shots increases to 4, 8 and 16,
the fine-tuned variants of Proto-CLIP outperform Tip-F. Proto-CLIP-$F$-$Q^{T}$
performs better than Proto-CLIP-$F$ on most datasets by using the data
augmentation of query images during training.

Dataset | ImageNet | FGVC | Pets | Cars | EuroSAT | Caltech101 | SUN397 | DTD | Flowers | Food101 | UCF101 | FewSOL
---|---|---|---|---|---|---|---|---|---|---|---|---
# classes | 1,000 | 100 | 37 | 196 | 10 | 100 | 397 | 47 | 102 | 101 | 101 | 52
Zero-shot CLIP [1] | 60.33 | 17.10 | 85.83 | 55.74 | 37.52 | 85.92 | 58.52 | 42.20 | 66.02 | 77.32 | 61.35 | 25.91
1 shots
Linear-Probe CLIP [1] | 22.07 | 12.89 | 30.14 | 24.64 | 51.00 | 70.62 | 32.80 | 29.59 | 58.07 | 30.13 | 41.43 | -
CoOp [8] | 57.15 | 9.64 | 85.89 | 55.59 | 50.63 | 87.53 | 60.29 | 44.39 | 68.12 | 74.32 | 61.92 | -
CLIP-A [9] | 61.20 | 17.49 | 85.99 | 55.13 | 61.40 | 88.60 | 61.30 | 45.80 | 73.49 | 76.82 | 62.20 | -
Tip [10] | 60.70 | 19.05 | 86.10 | 57.54 | 54.38 | 87.18 | 61.30 | 46.22 | 73.12 | 77.42 | 62.60 | 27.30
Tip-F [10] | 61.13 | 20.22 | 87.00 | 58.86 | 59.53 | 89.33 | 62.50 | 49.65 | 79.98 | 77.51 | 64.87 | 27.91
Proto-CLIP | 60.31 | 19.59 | 86.10 | 57.29 | 55.53 | 87.99 | 60.81 | 46.04 | 76.98 | 77.36 | 63.15 | 27.09
Proto-CLIP-$F$ | 60.32 | 19.50 | 85.72 | 57.34 | 54.93 | 88.07 | 60.83 | 35.64 | 77.47 | 77.34 | 63.07 | 22.22
Proto-CLIP-$F$-${Q^{T}}$ | 59.12 | 16.26 | 83.62 | 52.77 | 61.95 | 88.48 | 61.43 | 32.27 | 68.53 | 75.16 | 62.44 | 21.65
2 shots
Linear-Probe CLIP [1] | 31.95 | 17.85 | 43.47 | 36.53 | 61.58 | 78.72 | 44.44 | 39.48 | 73.35 | 42.79 | 53.55 | -
CoOp [8] | 57.81 | 18.68 | 82.64 | 58.28 | 61.50 | 87.93 | 59.48 | 45.15 | 77.51 | 72.49 | 64.09 | -
CLIP-A [9] | 61.52 | 20.10 | 86.73 | 58.74 | 63.90 | 89.37 | 63.29 | 51.48 | 81.61 | 77.22 | 67.12 | -
Tip [10] | 60.96 | 21.21 | 87.03 | 57.93 | 61.68 | 88.44 | 62.70 | 49.47 | 79.13 | 77.52 | 64.74 | 26.22
Tip-F [10] | 61.69 | 23.19 | 87.03 | 61.50 | 66.15 | 89.74 | 63.64 | 53.72 | 82.30 | 77.81 | 66.43 | 27.43
Proto-CLIP | 60.64 | 22.14 | 87.38 | 60.01 | 64.89 | 89.05 | 63.12 | 51.06 | 83.39 | 77.34 | 67.46 | 28.35
Proto-CLIP-$F$ | 60.64 | 22.14 | 87.38 | 60.04 | 64.86 | 89.09 | 63.20 | 49.88 | 83.52 | 77.34 | 67.49 | 26.17
Proto-CLIP-$F$-${Q^{T}}$ | 60.48 | 20.01 | 85.28 | 60.02 | 63.59 | 89.49 | 65.46 | 45.69 | 81.20 | 76.15 | 68.83 | 25.91
4 shots
Linear-Probe CLIP [1] | 41.29 | 23.57 | 56.35 | 48.42 | 68.27 | 84.34 | 54.59 | 50.06 | 84.80 | 55.15 | 62.23 | -
CoOp [8] | 59.99 | 21.87 | 86.70 | 62.62 | 70.18 | 89.55 | 63.47 | 53.49 | 86.20 | 73.33 | 67.03 | -
CLIP-A [9] | 61.84 | 22.59 | 87.46 | 62.45 | 73.38 | 89.98 | 65.96 | 56.86 | 87.17 | 77.92 | 69.05 | -
Tip [10] | 60.98 | 22.41 | 86.45 | 61.45 | 65.32 | 89.39 | 64.15 | 53.96 | 83.80 | 77.54 | 66.46 | 28.70
Tip-F [10] | 62.52 | 25.80 | 87.54 | 64.57 | 74.12 | 90.56 | 66.21 | 57.39 | 88.83 | 78.24 | 70.55 | 29.13
Proto-CLIP | 61.30 | 23.25 | 87.19 | 63.33 | 68.67 | 89.57 | 65.51 | 55.91 | 88.23 | 77.58 | 69.50 | 29.13
Proto-CLIP-$F$ | 61.30 | 23.31 | 86.95 | 63.34 | 68.52 | 89.62 | 65.57 | 57.21 | 88.27 | 77.58 | 69.55 | 27.09
Proto-CLIP-$F$-${Q^{T}}$ | 61.80 | 27.63 | 87.11 | 66.24 | 80.64 | 91.81 | 68.09 | 56.86 | 89.85 | 76.94 | 70.16 | 30.30
8 shots
Linear-Probe CLIP [1] | 49.55 | 29.55 | 65.94 | 60.82 | 76.93 | 87.78 | 62.17 | 56.56 | 92.00 | 63.82 | 69.64 | -
CoOp [8] | 61.56 | 26.13 | 85.32 | 68.43 | 76.73 | 90.21 | 65.52 | 59.97 | 91.18 | 71.82 | 71.94 | -
CLIP-A [9] | 62.68 | 26.25 | 87.65 | 67.89 | 77.93 | 91.40 | 67.50 | 61.00 | 91.72 | 78.04 | 73.30 | -
Tip [10] | 61.45 | 25.59 | 87.03 | 62.93 | 67.95 | 89.83 | 65.62 | 58.63 | 87.98 | 77.76 | 68.68 | 29.22
Tip-F [10] | 64.00 | 30.21 | 88.09 | 69.25 | 77.93 | 91.44 | 68.87 | 62.71 | 91.51 | 78.64 | 74.25 | 32.43
Proto-CLIP | 62.12 | 27.63 | 88.04 | 64.93 | 69.42 | 90.22 | 67.37 | 59.34 | 92.08 | 77.90 | 71.08 | 29.83
Proto-CLIP-$F$ | 63.92 | 31.32 | 88.55 | 70.35 | 78.94 | 92.54 | 69.59 | 62.35 | 93.79 | 78.29 | 74.81 | 33.26
Proto-CLIP-$F$-${Q^{T}}$ | 64.03 | 35.82 | 87.46 | 71.50 | 81.89 | 92.62 | 70.02 | 64.01 | 94.28 | 78.61 | 75.34 | 32.70
16 shots
Linear-Probe CLIP [1] | 55.87 | 36.39 | 76.42 | 70.08 | 82.76 | 90.63 | 67.15 | 63.97 | 94.95 | 70.17 | 73.72 | -
CoOp [8] | 62.95 | 31.26 | 87.01 | 73.36 | 83.53 | 91.83 | 69.26 | 63.58 | 94.51 | 74.67 | 75.71 | -
CLIP-A [9] | 63.59 | 32.10 | 87.84 | 74.01 | 84.43 | 92.49 | 69.55 | 65.96 | 93.90 | 78.25 | 76.76 | -
Tip [10] | 62.02 | 29.76 | 88.14 | 66.77 | 70.54 | 90.18 | 66.85 | 60.93 | 89.89 | 77.83 | 70.58 | 28.87
Tip-F [10] | 65.51 | 35.55 | 89.70 | 75.74 | 84.54 | 92.86 | 71.47 | 66.55 | 94.80 | 79.43 | 78.03 | 34.04
Proto-CLIP | 62.77 | 29.67 | 88.61 | 68.11 | 72.95 | 91.08 | 68.09 | 61.64 | 92.94 | 78.11 | 73.35 | 29.96
Proto-CLIP-$F$ | 65.75 | 37.56 | 89.62 | 75.25 | 83.53 | 93.43 | 71.94 | 68.56 | 95.78 | 79.09 | 77.50 | 35.22
Proto-CLIP-$F$-${Q^{T}}$ | 65.91 | 40.65 | 89.34 | 76.76 | 86.59 | 93.59 | 72.19 | 68.50 | 96.35 | 79.34 | 78.11 | 34.70

Table 5: Few-shot classification results on different datasets using the
ResNet50 backbone.

###  4.3 Real World Experiments

Figure 5: Results for the real world setup with top-5 predictions from the
Proto-CLIP-$F$ (ViT-L/14) model trained on FewSOL-198 [15]. The Speech-To-Text
is performed via Whisper [35].

As an application, we have built a robotic system to verify the effectiveness
of Proto-CLIP for object recognition in the real world. Fig. 5 illustrates our
pipeline for the system. It takes human instruction in the form of voice
commands as input such as “pick something” or “grasp something”. The system
first applies Automatic Speech Recognition (ASR) to convert voice input to
text using OpenAI Whisper [35]. Then the system grounds the noun in the human
instruction into a target object observed from an input image. This is
achieved by joint object segmentation and classification. We utilize unseen
object instance segmentation [36] to segment objects in cluttered scenes and
then classify each segmented object with Proto-CLIP. By matching the noun with
the class labels, the system can ground the target in the image. Once the
target object is recognized, we use Contact-GraspNet [37] for grasp planning
and MoveIt motion planning toolbox [38] to pick and place the target. See the
supplementary material for more real-world results.

##  5 Limitations

Proto-CLIP performs poorly in low-shot regimes, as is evident from Table 5. A
hyperparameter grid search is necessary for each new dataset, following the
methodology of Tip-Adapter. This requirement applies to every combination of
the new dataset and the backbone. Embracing the diversity of datasets, our
system thrives on the need for different set-ups. When encountering a new
dataset, we actively compare the effectiveness of $F$ and $F$-$Q^{T}$ to
determine the optimal choice. This dynamic approach transforms the potential
weakness into a strength, allowing us to adapt and maximize performance for
every unique dataset. During our observations, we discovered that data
transformations play a crucial role in building the cache model.

##  6 Conclusion and Future Work

We have introduced a novel method for few-shot learning based on the CLIP
vision-language model. Our method learns image prototypes and text prototypes
from few-shot training examples and aligns the corresponding image-text
prototypes for classification. The model is equipped with learnable image
memory and text memory for support images and a learnable adapter for query
images. Compared to previous CLIP-based few-shot learning methods, our method
is flexible in configuring these learnable components, resulting in powerful
learned models.

Good feature representation is the key in few-shot learning. Future work
includes how to further improve feature representation learning compared to
CLIP models. One idea is to adapt more powerful vision-language models such as
GPT variants. The FewSOL dataset also provides multiview and depth information
about objects. Exploring this 3D information in few-shot object recognition is
also a promising direction.

#### Acknowledgments

This work was supported in part by the DARPA Perceptually-enabled Task
Guidance (PTG) Program under contract number HR00112220005.

## References

  * Radford et al. [2021] A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, et al.  Learning transferable visual models from natural language supervision.  In _International Conference on Machine Learning_ , pages 8748–8763, 2021.
  * Wang et al. [2020] Y. Wang, Q. Yao, J. T. Kwok, and L. M. Ni.  Generalizing from a few examples: A survey on few-shot learning.  _ACM computing surveys (CSUR)_ , 53(3):1–34, 2020\.
  * Calli et al. [2015] B. Calli, A. Walsman, A. Singh, S. Srinivasa, P. Abbeel, and A. M. Dollar.  Benchmarking in manipulation research: The YCB object and model set and benchmarking protocols.  _arXiv preprint arXiv:1502.03143_ , 2015.
  * Lin et al. [2014] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, and C. L. Zitnick.  Microsoft COCO: Common objects in context.  In _European Conference on Computer Vision (ECCV)_ , pages 740–755. Springer, 2014.
  * Deng et al. [2009] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei.  ImageNet: A large-scale hierarchical image database.  In _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , pages 248–255, 2009.
  * Krishna et al. [2017] R. Krishna, Y. Zhu, O. Groth, J. Johnson, K. Hata, J. Kravitz, S. Chen, Y. Kalantidis, L.-J. Li, et al.  Visual genome: Connecting language and vision using crowdsourced dense image annotations.  _International Journal of Computer Vision (IJCV)_ , 123(1):32–73, 2017.
  * Tian et al. [2020] Y. Tian, Y. Wang, D. Krishnan, J. B. Tenenbaum, and P. Isola.  Rethinking few-shot image classification: a good embedding is all you need?  In _Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XIV 16_ , pages 266–282. Springer, 2020.
  * Zhou et al. [2022] K. Zhou, J. Yang, C. C. Loy, and Z. Liu.  Learning to prompt for vision-language models.  _International Journal of Computer Vision_ , 130(9):2337–2348, 2022.
  * Gao et al. [2021] P. Gao, S. Geng, R. Zhang, T. Ma, R. Fang, Y. Zhang, H. Li, and Y. Qiao.  Clip-adapter: Better vision-language models with feature adapters.  _arXiv 2110.04544_ , 2021.
  * Zhang et al. [2022] R. Zhang, Z. Wei, R. Fang, P. Gao, K. Li, J. Dai, Y. Qiao, and H. Li.  Tip-adapter: Training-free adaption of clip for few-shot classification.  _arXiv preprint arXiv:2207.09519_ , 2022.
  * Finn et al. [2017] C. Finn, P. Abbeel, and S. Levine.  Model-agnostic meta-learning for fast adaptation of deep networks.  In _International Conference on Machine Learning (ICML)_ , pages 1126–1135, 2017.
  * Snell et al. [2017] J. Snell, K. Swersky, and R. Zemel.  Prototypical networks for few-shot learning.  _Advances in Neural Information Processing Systems (NeurIPS)_ , 30, 2017.
  * Triantafillou et al. [2019] E. Triantafillou, T. Zhu, V. Dumoulin, P. Lamblin, U. Evci, K. Xu, R. Goroshin, C. Gelada, K. Swersky, P.-A. Manzagol, et al.  Meta-dataset: A dataset of datasets for learning to learn from few examples.  _arXiv preprint arXiv:1903.03096_ , 2019.
  * Doersch et al. [2020] C. Doersch, A. Gupta, and A. Zisserman.  Crosstransformers: spatially-aware few-shot transfer.  _Advances in Neural Information Processing Systems (NeurIPS)_ , 33:21981–21993, 2020.
  * P et al. [2022] J. P, Y.-W. Chao, and Y. Xiang.  Fewsol: A dataset for few-shot object learning in robotic environments.  _arXiv preprint arXiv:2207.03333_ , 2022.
  * Udandarao et al. [2022] V. Udandarao, A. Gupta, and S. Albanie.  Sus-x: Training-free name-only transfer of vision-language models.  _arXiv preprint arXiv:2211.16198_ , 2022.
  * Rombach et al. [2022] R. Rombach, A. Blattmann, D. Lorenz, P. Esser, and B. Ommer.  High-resolution image synthesis with latent diffusion models.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 10684–10695, 2022.
  * Gidaris and Komodakis [2018] S. Gidaris and N. Komodakis.  Dynamic few-shot visual learning without forgetting.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , pages 4367–4375, 2018.
  * Qi et al. [2018] H. Qi, M. Brown, and D. G. Lowe.  Low-shot learning with imprinted weights.  In _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , pages 5822–5830, 2018.
  * Chen et al. [2019] W.-Y. Chen, Y.-C. Liu, Z. Kira, Y.-C. F. Wang, and J.-B. Huang.  A closer look at few-shot classification.  _arXiv preprint arXiv:1904.04232_ , 2019.
  * Vinyals et al. [2016] O. Vinyals, C. Blundell, T. Lillicrap, D. Wierstra, et al.  Matching networks for one shot learning.  _Advances in Neural Information Processing Systems (NeurIPS)_ , 29, 2016.
  * Sung et al. [2018] F. Sung, Y. Yang, L. Zhang, T. Xiang, P. H. Torr, and T. M. Hospedales.  Learning to compare: Relation network for few-shot learning.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , pages 1199–1208, 2018.
  * Oord et al. [2018] A. v. d. Oord, Y. Li, and O. Vinyals.  Representation learning with contrastive predictive coding.  _arXiv preprint arXiv:1807.03748_ , 2018.
  * Krause et al. [2013] J. Krause, M. Stark, J. Deng, and L. Fei-Fei.  3d object representations for fine-grained categorization.  In _Proceedings of the IEEE international conference on computer vision workshops_ , pages 554–561, 2013.
  * Soomro et al. [2012] K. Soomro, A. R. Zamir, and M. Shah.  Ucf101: A dataset of 101 human actions classes from videos in the wild.  _arXiv preprint arXiv:1212.0402_ , 2012.
  * Fei-Fei et al. [2004] L. Fei-Fei, R. Fergus, and P. Perona.  Learning generative visual models from few training examples: An incremental bayesian approach tested on 101 object categories.  In _2004 conference on computer vision and pattern recognition workshop_ , pages 178–178. IEEE, 2004.
  * Nilsback and Zisserman [2008] M.-E. Nilsback and A. Zisserman.  Automated flower classification over a large number of classes.  In _2008 Sixth Indian Conference on Computer Vision, Graphics & Image Processing_, pages 722–729. IEEE, 2008.
  * Xiao et al. [2010] J. Xiao, J. Hays, K. A. Ehinger, A. Oliva, and A. Torralba.  Sun database: Large-scale scene recognition from abbey to zoo.  In _2010 IEEE computer society conference on computer vision and pattern recognition_ , pages 3485–3492. IEEE, 2010.
  * Cimpoi et al. [2014] M. Cimpoi, S. Maji, I. Kokkinos, S. Mohamed, and A. Vedaldi.  Describing textures in the wild.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 3606–3613, 2014.
  * Helber et al. [2019] P. Helber, B. Bischke, A. Dengel, and D. Borth.  Eurosat: A novel dataset and deep learning benchmark for land use and land cover classification.  _IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing_ , 12(7):2217–2226, 2019.
  * Maji et al. [2013] S. Maji, E. Rahtu, J. Kannala, M. Blaschko, and A. Vedaldi.  Fine-grained visual classification of aircraft.  _arXiv preprint arXiv:1306.5151_ , 2013.
  * Parkhi et al. [2012] O. M. Parkhi, A. Vedaldi, A. Zisserman, and C. Jawahar.  Cats and dogs.  In _2012 IEEE conference on computer vision and pattern recognition_ , pages 3498–3505. IEEE, 2012.
  * Bossard et al. [2014] L. Bossard, M. Guillaumin, and L. Van Gool.  Food-101–mining discriminative components with random forests.  In _Computer Vision–ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part VI 13_ , pages 446–461. Springer, 2014.
  * Van Der Maaten [2014] L. Van Der Maaten.  Accelerating t-sne using tree-based algorithms.  _The journal of machine learning research_ , 15(1):3221–3245, 2014.
  * Radford et al. [2022] A. Radford, J. W. Kim, T. Xu, G. Brockman, C. McLeavey, and I. Sutskever.  Robust speech recognition via large-scale weak supervision.  _arXiv preprint arXiv:2212.04356_ , 2022.
  * Lu et al. [2022] Y. Lu, Y. Chen, N. Ruozzi, and Y. Xiang.  Mean shift mask transformer for unseen object instance segmentation.  _arXiv preprint arXiv:2211.11679_ , 2022.
  * Sundermeyer et al. [2021] M. Sundermeyer, A. Mousavian, R. Triebel, and D. Fox.  Contact-graspnet: Efficient 6-dof grasp generation in cluttered scenes.  In _2021 IEEE International Conference on Robotics and Automation (ICRA)_ , pages 13438–13444. IEEE, 2021.
  * Chitta et al. [2012] S. Chitta, I. Sucan, and S. Cousins.  Moveit![ros topics].  _IEEE Robotics & Automation Magazine_, 19(1):18–19, 2012.


# Copositivity for a class of fourth order symmetric tensors ††thanks: This
work was supported by the National Natural Science Foundation of P.R. China
(Grant No. 11571095, 11601134, 11701154).

Xudong Li , Yisheng Song School of Mathematics and Information Science, Henan
Normal University, XinXiang HeNan, P.R. China, 453007. Email:
<EMAIL_ADDRESS>author. School of Mathematical Sciences,
Chongqing Normal University, Chongqing, P.R. China, 401331. Email:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we mainly discuss the analytic expression of exact copositivity
of 4th order symmetric tensor defined by the special physical model. We first
show that for the general 4th order 2-dimensional symmetric tensor, it can be
transformed into solving the quadratic polynomials, and then we give a
necessary and sufficient condition to test the copositivity of 4th order
2-dimensional symmetric tensor. Based on this, we consider a special 4th order
3-dimensional symmetric tensor defined by the vacuum stability for
$\mathbb{Z}_{3}$ scalar dark matter, and obtain the necessary and sufficient
condition for its copositivity.

Key words: 4th order Tensors, Symmetric tensors, Homogeneous polynomial,
Copositive tensors.

AMS subject classifications (2010): 47H60, 34B10, 47A52, 47J10, 15A48

##  1 Introduction

The vacuum stability of the standard model with multiple real scalar fields or
multiple microscopic particle potential energy is one of the hot research
fields at the forefront of science and technology. When this type of problem
is transformed into a mathematical model, it is a kind of homogeneous
polynomial optimization problem. For example, multiple real scalar field
potential energies in particle physics, scalar dark matter potential energies
and other hot research fields are specific potential energy models, which are
4 degree homogeneous polynomials in the real number domain or the complex
number domain. Recently, Kannike [15, 16, 17] studied the vacuum stability of
general scalar potentials of a few fields. The most general scalar quartic
potential of the SM Higgs $H_{1}$, an inert doublet $H_{2}$ and a complex
singlet $S$ which is symmetric under $\mathbb{Z}_{3}$ group is

$V(h_{1},h_{2},s)=V(\phi)=\mathcal{V}\phi^{4}=\sum\limits_{i,j,k,l=1}^{3}v_{ijkl}\phi_{i}\phi_{j}\phi_{k}\phi_{l}$
(1.1)

where $\phi=(\phi_{1},\phi_{2},\phi_{3})^{T}=(h_{1},h_{2},s)^{T}$,
$h_{1}=|H_{1}|$, $h_{2}=|H_{2}|$, $H_{2}^{\dagger}H_{1}=h_{1}h_{2}\rho
e^{i\phi}$, $S=se^{i\phi_{S}}$, $\mathcal{V}=(v_{ijkl})$ is a 4th order
3-dimensional real symmetric tensor. In order to determine the vacuum
stability of the physical model, it is necessary to obtain the precise
expression of the corresponding quartic polynomial optimization problem.

It is obvious that the vacuum stability for $\mathbb{Z}_{3}$ scalar dark
matter $V(h_{1},h_{2},s)$ is really equivalent to the (strict) copositivity of
the tensor $\mathcal{V}=(v_{ijkl})$. The concept of (strict) copositivity and
(semi)positive definite of symmetric tensor was first introduced by Qi [23,
24] in 2013. An mth order n-dimensional real tensor
$\mathcal{A}=(a_{i_{1}{i_{2}}\ldots{i_{m}}})$ is said to be

(i) semipositive definite if $\mathcal{A}x^{m}\geq 0$ for all x
$\in\mathbb{R}^{n}$ and even number m;

(ii) positive definite if $\mathcal{A}x^{m}>0$ for all x
$\in\mathbb{R}^{n}\backslash\\{0\\}$ and even number m;

(iii) copositive if $\mathcal{A}x^{m}\geq 0$ for all x
$\in\mathbb{R}_{+}^{n}$;

(iv) strictly copositive if $\mathcal{A}x^{m}>0$ for all x
$\in\mathbb{R}_{+}^{n}\backslash\\{0\\}$.

Song-Qi [28] proposed a method to test the (strict) copositivity of symmetric
tensors by using principal sub-tensors, and gave the equivalent definition of
(strictly) copositive tensors in the sense of any norm on $\mathbb{R}^{n}$.
Song-Qi [33] introduced the concepts of Pareto $H$-eigenvalue and
$Z$-eigenvalue in higher order symmetric tensors by means of Lagrange
multipliers, and gave the relation between the Pareto $H$-eigenvalue
($Z$-eigenvalue) and the (strict) copositivity of corresponding tensor. Qi
[24] proved a symmetric tensor is (strictly) copositive if each sum of a
diagonal element and all the negative off-diagonal elements in the same row is
(positive) nonegative. Song-Qi [31] presented that a symmetric tensor is
(strictly) copositive if and only if it is (strictly) semipositive.
$\mathcal{A}$ said to be (strictly) semipositive iff for each $x\geq 0$ and
$x\neq 0$, there exists an index $k\in{1,2,\ldots,n}$ such that

$x_{k}>0$ and $(\mathcal{A}x^{m-1})_{k}\geq 0(>0)$

The notion is firstly used by Song-Qi [29]. This kind of tensors has good
properties in the study of tensor complementary problem. Song-Qi [29] proved
every strictly semi-positive tensor is a $Q$-tensor, in other words, if
$\mathcal{A}$ is a strictly semi-positive tensor, then the complementarity
problem $(q,\mathcal{A})$ must have a solution for all $q\in\mathbb{R}^{n}$.
Furthermore, we can use it to explore the copositivity of tensors and its
applications [30, 34, 36, 38]. More questions about TCP can be find in ref.
[1, 2, 3, 4, 8, 9, 10, 14, 20, 39, 40, 41] .

Besides, some alternative numerical algorithms for copositivity of tensors
have been proposed. Chen-Huang-Qi [5] studied some basic theories of
copositivity detection of symmetric tensor and presented corresponding
numerical algorithms. Li-Zhang-Huang-Qi [18] proposed an SDP relaxation
algorithm to test the copositivity of higher order tensors. The more numerical
algorithms for checking copositivity of high order tensors were presented by
Chen-Huang-Qi [6], Chen-Wang [7]. For more details about copositivity
algorithms, also see [21, 25, 27]. But these algorithms are not designed for
the 4th order symmetric tensor, and obviously can not get the exact analytical
conditions required by the practical problems in the physical background.

Recently, Song-Qi [32] and Liu-Song [19] have presented different sufficient
conditions of copositivity for 4th order 3-dimensional symmetric tensors to
prove the vacuum stability conditions. However, the necessary and sufficient
conditions for copositivity have not been obtained. Guo [11] showed a
necessary and sufficient condition for the positive definite problem of a
binary quartic form, it is noteworthy that the conclusion in this paper is to
use the characteristics of conic curve rather than the discriminant. Very
recently, Qi-Song-Zhang [26] showed new necessary and sufficient conditions
for quartic polynomial to be positive for all positive reals. Song [37]
presented the positive definiteness of 4th order symmetric tensor.

In this paper, we work on seeking analytically checable necessary and
sufficient condition for copositivity of 4th order symmetric tensor. Firstly,
motivated by Qi-Song-Zhang’s [26] result, we first show a simple analytical
expression of copositivity of 4th order 2-dimensional symmetric tensor.
Furthermore, with the help of these conclusions, we discuss copositivity of a
4th order 3-dimensional symmetric tensor defined by vacuum stability for
$\mathbb{Z}_{3}$ scalar dark matter.

##  2 Preliminaries and Basic facts

In the whole paper, we use $x^{T}$ to represent the transpose of $x$. Let
$I_{n}:=\\{1,2,\ldots,n\\}$, then
$\mathbb{R}^{n}$:=$\\{(x_{1},x_{2},\ldots,x_{n})^{T}$$\mid$
$x_{i}\in\mathbb{R}$, $i\in I_{n}\\}$,
$\mathbb{R}^{n}_{+}:=\\{x\in\mathbb{R}^{n}|x\geq 0\\}$,
$\mathbb{R}^{n}_{++}:=\\{x\in\mathbb{R}^{n}|x>0\\}$, where $\mathbb{R}$
represents the whole set of real numbers, and $x\geq 0$ $(x>0)$ means that
$x_{i}\geq 0$ $(x_{i}>0)$ for all $i\in I_{n}$. The tensors discussed in this
article are all real. Let $\mathcal{A}=(a_{i_{1}i_{2}\ldots i_{m}})$ be an
m-order n-dimensional real tensor where $i_{j}\in I_{n}$, $j\in I_{m}$.
Suppose $\sigma$ is an arbitrary transformation, then we call $\mathcal{A}$ a
symmetric tensor if and only if

$a_{\sigma(i_{1},i_{2},\ldots,i_{m})}=a_{i_{1}i_{2}\cdots i_{m}}$,

the set of all symmetric tensors is denoted as $S_{m,n}$. Furthermore, for any
$x\in\mathbb{R}^{n}$, $\mathcal{A}x^{m-1}$ is an n-dimensional real vector,
and its $i$th element is represented as

$(\mathcal{A}x^{m-1})_{i}:=\sum^{n}_{i_{2},\ldots,i_{m}=1}a_{ii_{2}\ldots
i_{m}}x_{i_{2}}\cdots x_{i_{m}}$.

Then it is easy to get that $\mathcal{A}x^{m}$ is a homogeneous polynomial of
degree n, which can be defined as

$\mathcal{A}x^{m}:=x^{T}(\mathcal{A}x^{m-1})=\sum^{n}_{i_{1}\cdots
i_{m}=1}a_{i_{1}\cdots i_{m}}x_{i_{1}}\cdots x_{i_{m}}$.

The physical model of the potential energy vacuum stability that this article
relies on determines a fourth-order three-dimensional symmetric tensor. We can
easily find that a 4th order 3-dimensional real tensor $\mathcal{A}$ contains
81 entries in the real field, if it is symmetric, it only contains 15
elements, which greatly simplifies the complexity of studying this practical
problem.

For an $n\times n$ matrix, when $n>4$, it is difficult to obtain the necessary
and sufficient conditions for its copositivity. However, for the low-
dimensional matrices with $n\leq 4$, there have been relatively simple methods
to discriminate. The method for judging the copositivity of 2-dimensional
matrices given below are well known.

######  Lemma 2.1.

[13, 22] Let $\mathbf{A}=(a_{ij})$ be a $2\times 2$ real symmetric matrix,
then $\mathbf{A}$ is (strictly) copositive if and only if

$a_{11}\geq 0(>0)$, $a_{22}\geq 0(>0)$, $a_{12}+\sqrt{a_{11}a_{22}}\geq
0(>0)$.

Let $\|\cdot\|$ be any norm in Euclidean space, after limiting the norm of $x$
to 1, then Song and Qi presented the equivalent definition of copositivity and
strict copositivity of any symmetric tensors.

######  Lemma 2.2.

([23]) Let $\mathcal{A}$ be a real mth order n-dimensional symmetric tensor.
Then

(i) $\mathcal{A}$ is copositive if and only if $\mathcal{A}x^{m}\geq 0$ for
all $x\in\mathbb{R}_{+}^{n}$ with $\|x\|=1$;

(ii) $\mathcal{A}$ is strictly copositive if and only if $\mathcal{A}x^{m}>0$
for all $x\in\mathbb{R}_{+}^{n}$ with $\|x\|=1$.

The image and properties of the quadratic function have been studied a long
time ago. The following are the necessary and sufficient conditions for the
quadratic function to be always greater than zero.

######  Lemma 2.3.

Let f(t) be a quadratic polynomial defined as follows

$f(t)=at^{2}+bt+c,$

where $a>0$. Then $f(t)>0(\geq 0)$ for all $t\geq 0$ if and only if

(1)$b\geq 0$ and $c>0(\geq 0)$; or

(2)$b<0$ and $4ac-b^{2}>0(\geq 0)$

As early as 1994, Ulrich and Watson [12] proposed the necessary and sufficient
condition that the quartic polynomial is non-negative for all positive real
numbers. Recently, Qi-Song-Zhang [26] reexpressed their conclusions, and came
up with a simpler necessary and sufficient condition.

######  Lemma 2.4.

([26]) Let f(t) be a quartic polynomial defined as follows

$f(t)=at^{4}+bt^{3}+ct^{2}+dt+e,$

where $a>0$, $e>0$. Then $f(t)\geq 0$ for all $t>0$ if and only if

(1)$\Delta\leq 0$ and $b\sqrt{e}+d\sqrt{a}>0;$ or

(2)$b\geq 0$, $d\geq 0$ and $2\sqrt{ae}+c\geq 0$; or

(3)$\Delta\geq 0$, $|b\sqrt{e}-d\sqrt{a}|\leq 4\sqrt{ace+2ae\sqrt{ae}}$ and
either

(i)$-2\sqrt{ae}\leq c\leq 6\sqrt{ae}$, or

(ii)$c-6\sqrt{ae}>0$ and $b\sqrt{e}+d\sqrt{a}\geq-4\sqrt{ace-2ae\sqrt{ae}}$.

where $\Delta=4(12ae-3bd+c^{2})^{3}-(72ace+9bcd-2c^{3}-27ad^{2}-27b^{2}e)^{2}$

##  3 Main Results

The specific potential energy model determined by the vacuum stability of the
dark matter potential energy under the $\mathbb{Z}_{3}$ discrete group is a
quartic polynomial in the real number domain, and the quartic polynomial has a
one-to-one correspondence with the fourth-order symmetric tensor. Then the
mathematical model is transformed into a problem that the minimum value of a
homogeneous polynomial of degree 4 under the constraint of a non-negative
spherical surface is positive. It is also equivalent to the (strict)
copositivity of the 4th-order symmetric tensor.

###  3.1 4th order 2 dimensional symmetric tensors

In this part, we mainly discuss the analytical expression of the copositivity
of the low-dimensional 4th order tensor. Let $\mathcal{A}$ be a 4th order
2-dimensional symmetric tensor, then for a vector $x=(x_{1},x_{2})^{T}$,

$\displaystyle\begin{split}\mathcal{A}x^{4}&=\sum_{i,j,k,l=1}^{2}a_{ijkl}x_{i}x_{j}x_{k}x_{l}\\\
&=a_{1111}x_{1}^{4}+4a_{1112}x_{1}^{3}x_{2}+6a_{1122}x_{1}^{2}x_{2}^{2}+4a_{1222}x_{1}x_{2}^{3}+a_{2222}x_{2}^{4}\end{split}$
(3.1)

Obviously, if the tensor $\mathcal{A}$ is positive (semi-)definite, then it
must be strictly copositive (copositive). This gives the following two
theorems.

######  Theorem 3.1.

Let $\mathcal{A}$ be a 4th order 2-dimensional symmetric tensor with
$a_{1111}>0$, $a_{2222}>0$, then for any vector $x\in\mathbb{R}_{+}^{2}$, if

$I^{3}-27J^{2}\geq 0$,
$|a_{1222}\sqrt{a_{1111}}-a_{1112}\sqrt{a_{2222}}|\leq\sqrt{6a_{2222}a_{1122}a_{1111}+2a_{1111}a_{2222}\sqrt{a_{1111}a_{2222}}}$,

(i) $-\sqrt{a_{1111}a_{2222}}\leq 3a_{1122}\leq 3\sqrt{a_{1111}a_{2222}}$ or

(ii) $a_{1122}>\sqrt{a_{1111}a_{2222}}$,

$|a_{1222}\sqrt{a_{1111}}+a_{1112}\sqrt{a_{2222}}|\leq\sqrt{6a_{2222}a_{1122}a_{1111}-2a_{1111}a_{2222}\sqrt{a_{1111}a_{2222}}}$,

where

$I=a_{1111}a_{2222}-4a_{1112}a_{1222}+3a_{1221}^{2}$,

$J=a_{1111}a_{1122}a_{2222}+2a_{1112}a_{1122}a_{1222}-a_{1122}^{2}-a_{1111}a_{1222}^{2}-a_{1112}^{2}a_{2222}$.

then $\mathcal{A}$ is copositive.

###### Proof.

From lemma 2.2, let a vector $x=(x_{1},x_{2})^{T}$ be on the non-negative
sphere, i.e.

$\|x\|=\sqrt{x_{1}^{2}+x_{2}^{2}}=1$ and $x_{i}\in\mathbb{R}_{+}$ for $i=1,2$.

Suppose $x_{1}=0$, $x_{2}\neq 0$, then we can get
$\mathcal{A}x^{4}=a_{2222}x_{2}^{4}\geq 0$;
Suppose $x_{2}=0$, $x_{1}\neq 0$, then we can get
$\mathcal{A}x^{4}=a_{1111}x_{1}^{4}\geq 0$;
Suppose $x_{1}\neq 0$, $x_{2}\neq 0$, without loss of generality, we can
devide by $x_{1}^{4}$ on both sides of the equation, then $\mathcal{A}x^{4}$
can be rewritten as follows,

$\displaystyle\frac{\mathcal{A}x^{4}}{x_{1}^{4}}=a_{1111}+4a_{1112}\displaystyle\frac{x_{2}}{x_{1}}+6a_{1122}(\displaystyle\frac{x_{2}}{x_{1}})^{2}+4a_{1222}(\displaystyle\frac{x_{2}}{x_{1}})^{3}+a_{2222}(\displaystyle\frac{x_{2}}{x_{1}})^{4}$

Let $t=\displaystyle\frac{x_{2}}{x_{1}}$, $a=a_{2222}$, $b=4a_{1222}$,
$c=6a_{1122}$, $d=4a_{1112}$, $e=a_{1111}$, then

$f(t)=at^{4}+bt^{3}+ct^{2}+dt+e.$

In this way, if for any $t\in\mathbb{R}$, $f(t)\geq 0$, then
$\mathcal{A}x^{4}\geq 0$, that is, $\mathcal{A}$ is copositive. Therefore, as
long as the condition for determining the non-negative polynomial of degree 4
is given, the symmetric tensor $\mathcal{A}$ can be obtained as copositive.
Very naturally, from the theorem 3.2 of [35], we have $f(t)\geq 0$ for all
$t\geq 0$. That is, $\mathcal{A}$ is copositive. ∎

In the same way, using theorem 3.1 in [35], we can also have $f(t)>0$, and
hence, the following conclusion can be obtained easily, which is the
copositivity of $\mathcal{A}$ is proved as required.

######  Theorem 3.2.

Let $\mathcal{A}$ be a 4th order 2-dimensional symmetric tensor with
$a_{1111}>0$, $a_{2222}>0$, then for any vector
$x\in\mathbb{R}_{+}^{2}\setminus\\{0\\}$, if

(1) $I^{3}-27J^{2}=0$, $a_{1222}\sqrt{a_{1111}}=a_{1112}\sqrt{a_{2222}}$,

$a_{1222}^{2}+2a_{2222}\sqrt{a_{1111}a_{2222}}=6a_{1122}a_{2222}<6a_{2222}\sqrt{a_{1111}a_{2222}}$.

(2) $I^{3}-27J^{2}>0$,
$|a_{1222}\sqrt{a_{1111}}-a_{1112}\sqrt{a_{2222}}|\leq\sqrt{6a_{2222}a_{1122}a_{1111}+2a_{1111}a_{2222}\sqrt{a_{1111}a_{2222}}}$,

(i)$-\sqrt{a_{1111}a_{2222}}\leq 3a_{1122}\leq 3\sqrt{a_{1111}a_{2222}}$ or

(ii)$a_{1122}>\sqrt{a_{1111}a_{2222}}$,

$|a_{1222}\sqrt{a_{1111}}+a_{1112}\sqrt{a_{2222}}|\leq\sqrt{6a_{2222}a_{1122}a_{1111}-2a_{1111}a_{2222}\sqrt{a_{1111}a_{2222}}}$,

then $\mathcal{A}$ is strictly copositive.

With the help of lemma 2.4, now we give a necessary and sufficient condition
of copositivity of general 4th order 2-dimensional symmetric tensor.

######  Theorem 3.3.

Let $\mathcal{A}$ be a 4th order 2-dimensional symmetric tensor, then for any
vector $x\in\mathbb{R}_{+}^{2}$, $\mathcal{A}$ is copositive if and only if
$a_{1111}>0$, $a_{2222}>0$ and

(1) $I^{3}-27J^{2}\leq 0$,
$a_{1222}\sqrt{a_{1111}}+a_{1112}\sqrt{a_{2222}}>0$; or

(2) $a_{1222}\geq 0$, $a_{1112}\geq 0$, $3a_{1122}+\sqrt{a_{1111}a_{2222}}\geq
0$; or

(3) $I^{3}-27J^{2}\geq 0$,

$|a_{1112}\sqrt{a_{2222}}-a_{1222}\sqrt{a_{1111}}|\leq\sqrt{6a_{1111}a_{1122}a_{2222}+2a_{1111}a_{2222}\sqrt{a_{1111}a_{2222}}}$,

(i) $-\sqrt{a_{1111}a_{2222}}\leq 3a_{1122}\leq 3\sqrt{a_{1111}a_{2222}}$;

(ii) $a_{1122}>\sqrt{a_{1111}a_{2222}}$,

$a_{1112}\sqrt{a_{2222}}+a_{1222}\sqrt{a_{1111}}\geq-\sqrt{6a_{1111}a_{1122}a_{2222}-2a_{1111}a_{2222}\sqrt{a_{1111}a_{2222}}}$.

where

$I=a_{1111}a_{2222}-4a_{1112}a_{1222}+3a_{1221}^{2}$,

$J=a_{1111}a_{1122}a_{2222}+2a_{1112}a_{1122}a_{1222}-a_{1122}^{2}-a_{1111}a_{1222}^{2}-a_{1112}^{2}a_{2222}$.

###### Proof.

From lemma 2.2, let a vector $x=(x_{1},x_{2})^{T}$ be on the non-negative
sphere, i.e.

$\|x\|=\sqrt{x_{1}^{2}+x_{2}^{2}}=1$ and $x_{i}\in\mathbb{R}_{+}$ for i=1, 2.

Suppose $x_{1}=0$, $x_{2}\neq 0$, then we can get
$\mathcal{A}x^{4}=a_{2222}x_{2}^{4}\geq 0$;
Suppose $x_{2}=0$, $x_{2}\neq 0$, then we can get
$\mathcal{A}x^{4}=a_{1111}x_{1}^{4}\geq 0$;
Suppose $x_{1}\neq 0$, $x_{2}\neq 0$, without loss of generality, we can
rearrange the homogeneous polynomial $\mathcal{A}x^{4}$ as follows,

$\displaystyle\mathcal{A}x^{4}$
$\displaystyle=a_{1111}x_{1}^{4}+4a_{1112}x_{1}^{3}x_{2}+6a_{1122}x_{1}^{2}x_{2}^{2}+4a_{1222}x_{1}x_{2}^{3}+a_{2222}x_{2}^{4}$
$\displaystyle=x_{1}^{4}(a_{1111}+4a_{1112}\frac{x_{2}}{x_{1}}+6a_{1122}(\frac{x_{2}}{x_{1}})^{2}+4a_{1222}(\frac{x_{2}}{x_{1}})^{3}+a_{2222}(\frac{x_{2}}{x_{1}})^{4})$

we can devide by $x_{1}^{4}$ on both sides of the equation, then the
homogeneous polynomial $\mathcal{A}x^{4}$ can be rewritten as follows,

$\displaystyle\frac{\mathcal{A}x^{4}}{x_{1}^{4}}=a_{2222}(\frac{x_{2}}{x_{1}})^{4}+4a_{1222}(\frac{x_{2}}{x_{1}})^{3}+6a_{1122}(\frac{x_{2}}{x_{1}})^{2}+4a_{1112}\frac{x_{2}}{x_{1}}+a_{1111}.$

Obviously, for any $x\in\mathbb{R}_{++}^{2}$, $\mathcal{A}x^{4}\geq 0$ if and
only if

$f(t)=at^{4}+bt^{3}+ct^{2}+dt+e\geq 0,$

where $a=a_{2222},$ $b=4a_{1222},$ $c=6a_{1122},$ $d=4a_{1112},$ $e=a_{1111},$
$t=\displaystyle\frac{x_{2}}{x_{1}}>0$. Furthermore, from lemma 2.4, we have

$\displaystyle\Delta$
$\displaystyle=4(12ae-3bd+c^{2})^{3}-(72ace+9bcd-2c^{3}-27ad^{2}-27b^{2}e)^{2}$
$\displaystyle=4(12a_{1111}a_{2222}-12*4a_{1112}a_{1122}+12*3a_{1222}^{2})^{3}$
$\displaystyle\quad-(72*6a_{1111}a_{1122}a_{2222}+72*12a_{1112}a_{1122}a_{1222}-72*6a_{1122}^{3}$
$\displaystyle\quad-72*6a_{1112}^{2}a_{2222}-72*6a_{1111}a_{1122}^{2})^{2}$
$\displaystyle=4*12^{3}(a_{1111}a_{2222}-4a_{1112}a_{1122}+3a_{1222}^{2})^{3}$
$\displaystyle\quad-72^{2}*6^{2}(a_{1111}a_{1122}a_{2222}+2a_{1112}a_{1122}a_{1222}-a_{1122}^{3}-a_{1112}^{2}a_{2222}-a_{1111}a_{1122}^{2})^{2}$
$\displaystyle=4*12^{3}(I^{3}-27J^{2})$

where

$I=a_{1111}a_{2222}-4a_{1112}a_{1222}+3a_{1122}^{2}$,

$J=a_{1111}a_{1122}a_{2222}+2a_{1112}a_{1122}a_{1222}-a_{1122}^{2}-a_{1111}a_{1222}^{2}-a_{1112}^{2}a_{2222}$.

we can find that $\Delta$ and $I^{3}-27J^{3}$ have the same sign. In this way,
the problem of judging that there is $\mathcal{A}x^{4}\geq 0$ for all
$x\in\mathbb{R}_{++}$ is transformed into the problem of judging that there is
a quartic polynomial $f(t)\geq 0$ for all $t>0$. Then the conclusion can be
obtained by using lemma 2.4 and simple calculation directly. ∎

###  3.2 4th order 3 dimensional symmetric tensors

For a general 4th order 3-dimensional symmetric tensor, it is quite arduous to
obtain the judgement condition of its (strict) copositive, accordingly, we
consider starting from the specific problem.

Recently, Kannike [15, 16, 17] showed a physical example defined by vacuum
stability for $\mathbb{Z}_{3}$ scalar dark matter, In this subsection, we will
propose an analytical necessary and sufficient condition of copositivity for
the particle physical models. The most general scalar quartic of the SM Higgs
$H_{1}$, an inert doublet $H_{2}$, and a complex singlet $S$ can be written as

$\begin{split}V(h_{1},h_{2},s)&=\lambda_{1}|H_{1}|^{4}+\lambda_{2}|H_{2}|^{4}+\lambda_{3}|H_{1}|^{2}|H_{2}|^{2}+\lambda_{4}(H_{1}^{{\dagger}}H_{2})(H_{2}^{{\dagger}}H_{1})\\\
&\quad+\lambda_{S}|S|^{4}+\lambda_{S1}|S|^{2}|H_{1}|^{2}+\lambda_{S2}|S|^{2}|H_{2}|^{2}\\\
&\quad+\frac{1}{2}(\lambda_{S12}S^{2}H_{1}^{{\dagger}}H_{2}+\lambda_{S12}^{*}S^{{\dagger}2}H_{2}^{{\dagger}}H_{1})\\\
&=\lambda_{1}h_{1}^{4}+\lambda_{2}h_{2}^{4}+\lambda_{3}h_{1}^{2}h_{2}^{2}+\lambda_{4}\rho^{2}h_{1}^{2}h_{2}^{2}\\\
&\quad+\lambda_{S}s^{4}+\lambda_{S1}s^{2}h_{1}^{2}+\lambda_{S2}s^{2}h_{2}^{2}-|\lambda_{S12}|\rho
s^{2}h_{1}h_{2}\\\
&=\lambda_{S}s^{4}+M(h_{1},h_{2})s^{2}+\widetilde{V}(h_{1},h_{2})\end{split}$
(3.2)

where

$\qquad h_{1}=|H_{1}|,\quad h_{2}=|H_{2}|,\quad
H_{2}^{{\dagger}}H_{1}=h_{1}h_{2}\rho e^{i\phi},\quad
S=se^{i\phi_{S}},\quad\lambda_{S12}=-|\lambda_{S12}|,$ $\qquad
M(h_{1},h_{2})=\lambda_{S1}h_{1}^{2}+\lambda_{S2}h_{2}^{2}-|\lambda_{S12}|\rho
h_{1}h_{2},$ (3.3)
$\qquad\widetilde{V}(h_{1},h_{2})=\lambda_{1}h_{1}^{4}+\lambda_{2}h_{2}^{4}+\lambda_{3}h_{1}^{2}h_{2}^{2}+\lambda_{4}\rho^{2}h_{1}^{2}h_{2}^{2}.$
(3.4)

$\rho\in[0,1]$ of orbit space parameter implied by the Cauchy inequality
$0\leq|H_{1}^{{\dagger}}H_{2}|\leq|H_{1}||H_{2}|$. For any
$x=(x_{1},x_{2},x_{3})^{T}=(h_{1},h_{2},s)^{T}\in\mathbb{R}_{+}^{3}$. Let

$v_{1111}=\lambda_{1},$ $v_{2222}=\lambda_{2},$ $v_{3333}=\lambda_{S}$,

$v_{1122}=\displaystyle\frac{1}{6}(\lambda_{3}+\lambda_{4}\rho^{2}),$
$v_{1133}=\displaystyle\frac{1}{6}\lambda_{S1},$
$v_{2233}=\displaystyle\frac{1}{6}\lambda_{S2}$,

$v_{1233}=-\displaystyle\frac{1}{12}|\lambda_{S12}|\rho,$ $v_{ijkl}=0$ for
others,

then $V(h_{1},h_{2},s)=\mathcal{V}x^{4}$, where $\mathcal{V}=(v_{ijkl})$ is a
real symmetric tensor of 4th order 3-dimensional. As we all know, a general
4th-order 3-dimensional tensor should contain 81 elements, but it is obvious
that only 33 elements of the 3-dimensional tensor determined by the physical
model are not zero. In other words, the homogeneous polynomial
$\mathcal{V}x^{4}$ has only 7 terms, and the coefficients of the other 8 terms
are all $0$. Then, based on the special structure of the tensor $\mathcal{V}$,
the necessary and sufficient conditions for its copositivity are given. From
lemma 2.2, the following result is easy to get.

######  Theorem 3.4.

Let $\mathcal{V}=(v_{ijkl})$ be a real symmetric tensor of 4th order
3-dimensional given by (3.2) with $\lambda_{S}>0$. Then for any
$h=(h_{1},h_{2})\in\mathbb{R}^{2}_{+}\setminus\\{0\\},$ $\mathcal{V}$ is
(strictly) copositive if and only if

(1) $M(h_{1},h_{2})\geq 0,$ $\widetilde{V}(h_{1},h_{2})\geq 0(>0)$, or

(2) $M(h_{1},h_{2})<0,$
$4\lambda_{S}\widetilde{V}(h_{1},h_{2})-(M(h_{1},h_{2}))^{2}\geq 0(>0),$

where $M(h_{1},h_{2})$ and $\widetilde{V}(h_{1},h_{2})$ are respectively
defined by (3.3) and (3.4).

###### Proof.

For all $x=(h_{1},h_{2},s)^{T}\in\mathbb{R}^{3}_{+}\setminus\\{0\\}$, from the
equations (3.2), we can easily obtain that

$\mathcal{V}x^{4}=\lambda_{S}s^{4}+M(h_{1},h_{2})s^{2}+\widetilde{V}(h_{1},h_{2}),$

Let

$a=\lambda_{S},\ b=M(h_{1},h_{2}),\ c=\widetilde{V}(h_{1},h_{2}),\ t=s^{2}.$

then the homogeneous polynomial $\mathcal{V}x^{4}$ can be rewritten as

$f(t)=\lambda_{S}t^{2}+M(h_{1},h_{2})t+\widetilde{V}(h_{1},h_{2})$,

which can be seen as a one-variable quadratic polynomial about $t$, and the
desired conclusion can be directly obtain by lemma 2.3. ∎

It can be easily seen that $M(h_{1},h_{2})$ is a quadratic polynomial, hence,
it is clear that for all $h=(h_{1},h_{2})^{T}\in\mathbb{R}_{+}^{2}$,

$\displaystyle M(h_{1},h_{2})$
$\displaystyle=\lambda_{S1}h_{1}^{2}+\lambda_{S2}h_{2}^{2}-|\lambda_{S12}|\rho
h_{1}h_{2}$ $\displaystyle=\left(\begin{array}[]{cc}h_{1}&h_{2}\\\
\end{array}\right)\left(\begin{array}[]{cc}\lambda_{S1}&-\displaystyle\frac{1}{2}|\lambda_{S12}|\rho\vspace{1ex}\\\
-\displaystyle\frac{1}{2}|\lambda_{S12}|\rho&\lambda_{S2}\\\
\end{array}\right)\left(\begin{array}[]{c}h_{1}\vspace{2ex}\\\ h_{2}\\\
\end{array}\right).$

then $M(h_{1},h_{2})\geq 0$ is equivalent to the copositivity of its
coefficient matrix, from lemma 2.1, i.e.,

$\lambda_{S1}\geq 0$, $\lambda_{S2}\geq 0$,
$-\displaystyle\frac{1}{2}|\lambda_{S12}|\rho+\sqrt{\lambda_{S1}\lambda_{S2}}\geq
0$.

Let $x=(x_{1},x_{2})^{T}=(h_{1}^{2},h_{2}^{2})^{T}$, $x_{1}$,
$x_{2}\in\mathbb{R}_{+}$, we can rewrite $\widetilde{V}(h_{1},h_{2})$ as
follows,

$\displaystyle\widetilde{V}(h_{1},h_{2})$
$\displaystyle=\lambda_{1}h_{1}^{4}+\lambda_{2}h_{2}^{4}+\lambda_{3}h_{1}^{2}h_{2}^{2}+\lambda_{4}\rho^{2}h_{1}^{2}h_{2}^{2}$
$\displaystyle=\left(\begin{array}[]{cc}h_{1}^{2}&h_{2}^{2}\\\
\end{array}\right)\left(\begin{array}[]{cc}\lambda_{1}&\displaystyle\frac{1}{2}(\lambda_{3}+\lambda_{4}\rho^{2})\vspace{1ex}\\\
\displaystyle\frac{1}{2}(\lambda_{3}+\lambda_{4})\rho^{2}&\lambda_{2}\\\
\end{array}\right)\left(\begin{array}[]{c}h_{1}^{2}\vspace{2ex}\\\
h_{2}^{2}\\\ \end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}x_{1}&x_{2}\\\
\end{array}\right)\left(\begin{array}[]{cc}\lambda_{1}&\displaystyle\frac{1}{2}(\lambda_{3}+\lambda_{4}\rho^{2})\vspace{1ex}\\\
\displaystyle\frac{1}{2}(\lambda_{3}+\lambda_{4})\rho^{2}&\lambda_{2}\\\
\end{array}\right)\left(\begin{array}[]{c}x_{1}\vspace{2ex}\\\ x_{2}\\\
\end{array}\right).$

From lemma 2.1, we can get a similar conclusion easily. The copositivity of
$\widetilde{V}(h_{1},h_{2})$ is equivalent to

$\lambda_{1}\geq 0$, $\lambda_{2}\geq 0$,
$\displaystyle\frac{1}{2}(\lambda_{3}+\lambda_{4}\rho^{2})+\sqrt{\lambda_{1}\lambda_{2}}\geq
0$.

Similarly, $M(h_{1},h_{2})<0$ is equivalent to the strict copositivity of
$-M(h_{1},h_{2})$, and

$\displaystyle-M(h_{1},h_{2})$
$\displaystyle=-\lambda_{S1}h_{1}^{2}-\lambda_{S2}h_{2}^{2}+|\lambda_{S12}|\rho
h_{1}h_{2}$ $\displaystyle=\left(\begin{array}[]{cc}h_{1}&h_{2}\\\
\end{array}\right)\left(\begin{array}[]{cc}-\lambda_{S1}&\displaystyle\frac{1}{2}|\lambda_{S12}|\rho\vspace{1ex}\\\
\displaystyle\frac{1}{2}|\lambda_{S12}|\rho&-\lambda_{S2}\\\
\end{array}\right)\left(\begin{array}[]{c}h_{1}\vspace{2ex}\\\ h_{2}\\\
\end{array}\right)$

Then $M(h_{1},h_{2})<0$ if and only if

$\lambda_{S1}<0,\ \lambda_{S2}<0,\
\frac{1}{2}|\lambda_{S12}|\rho+\sqrt{\lambda_{S1}\lambda_{S2}}>0.$

It is obviously that
$\frac{1}{2}|\lambda_{S12}|\rho+\sqrt{\lambda_{S1}\lambda_{S2}}>0$ since
$\frac{1}{2}|\lambda_{S12}|\rho\geq 0$, and hence, $M(h_{1},h_{2})<0$ if and
only if

$\lambda_{S1}<0,\ \lambda_{S2}<0.$

Next in order to easily give the necessary and sufficient conditions for
$4\lambda_{S}\widetilde{V}(h_{1},h_{2})-(M(h_{1},h_{2}))^{2}\geq 0$, the
following variables are presented first,

$\displaystyle\quad\lambda_{40}^{{}^{\prime}}$
$\displaystyle=4\lambda_{S}\lambda_{1}-\lambda_{S1}^{2},\
\lambda_{04}^{{}^{\prime}}=4\lambda_{S}\lambda_{2}-\lambda_{S2}^{2},$
$\displaystyle\quad\lambda_{13}^{{}^{\prime}}$
$\displaystyle=2\lambda_{S2}|\lambda_{S12}|\rho,\
\lambda_{31}^{{}^{\prime}}=2\lambda_{S1}|\lambda_{S12}|\rho,$
$\displaystyle\quad\lambda_{22}^{{}^{\prime}}$
$\displaystyle=4\lambda_{S}\lambda_{3}+4\lambda_{S}\lambda_{4}\rho^{2}-|\lambda_{S12}|^{2}\rho^{2}-2\lambda_{S1}\lambda_{S2},$
$\displaystyle\quad\Delta^{{}^{\prime}}$
$\displaystyle=4(12\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}-3\lambda_{31}^{{}^{\prime}}\lambda_{13}^{{}^{\prime}}+\lambda_{22}^{{}^{\prime}})^{3}$
$\displaystyle-(72\lambda_{40}^{{}^{\prime}}\lambda_{22}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}+9\lambda_{31}^{{}^{\prime}}\lambda_{22}^{{}^{\prime}}\lambda_{13}^{{}^{\prime}}-2\lambda_{22}^{{}^{\prime}3}-27\lambda_{40}^{{}^{\prime}}\lambda_{13}^{{}^{\prime}2}-27\lambda_{31}^{{}^{\prime}2}\lambda_{04}^{{}^{\prime}})^{2}$

######  Proposition 3.5.

For any $h=(h_{1},h_{2})^{T}\in\mathbb{R}_{+}^{2}$,
$4\lambda_{S}\widetilde{V}(h_{1},h_{2})-(M(h_{1},h_{2}))^{2}\geq 0$ if and
only if $\lambda_{40}^{{}^{\prime}}>0,$ $\lambda_{04}^{{}^{\prime}}>0$ and

(1) $\Delta^{{}^{\prime}}\leq 0$,
$\lambda_{13}^{{}^{\prime}}\sqrt{\lambda_{40}^{{}^{\prime}}}+\lambda_{31}^{{}^{\prime}}\sqrt{\lambda_{04}^{{}^{\prime}}}>0$,
or

(2) $\lambda_{13}^{{}^{\prime}}\geq 0$, $\lambda_{31}^{{}^{\prime}}\geq 0$,
$\lambda_{22}^{{}^{\prime}}+2\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}\geq
0$, or

(3) $\Delta^{{}^{\prime}}\geq 0$,
$|\lambda_{31}^{{}^{\prime}}\sqrt{\lambda_{04}^{{}^{\prime}}}-\lambda_{13}^{{}^{\prime}}\sqrt{\lambda_{40}^{{}^{\prime}}}|\leq
4\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{22}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}+2\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}}$,

(i)
$-2\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}\leq\lambda_{22}^{{}^{\prime}}\leq
6\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}$;

(ii)
$\lambda_{22}^{{}^{\prime}}>6\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}$,

$\lambda_{31}^{{}^{\prime}}\sqrt{\lambda_{04}^{{}^{\prime}}}+\lambda_{13}^{{}^{\prime}}\sqrt{\lambda_{40}^{{}^{\prime}}}\geq-4\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{22}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}-2\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}}$.

###### Proof.

$\displaystyle\quad\widetilde{V}^{{}^{\prime}}(h_{1},h_{2})$
$\displaystyle=4\lambda_{S}\widetilde{V}(h_{1},h_{2})-(M(h_{1},h_{2}))^{2}$
$\displaystyle=4\lambda_{S}(\lambda_{1}h_{1}^{4}+\lambda_{2}h_{2}^{4}+\lambda_{3}h_{1}^{2}h_{2}^{2}+\lambda_{4}\rho^{2}h_{1}^{2}h_{2}^{2})$
$\displaystyle\quad-(\lambda_{S1}h_{1}^{2}+\lambda_{S2}h_{2}^{2}-|\lambda_{S12}|\rho
h_{1}h_{2})^{2}$
$\displaystyle=(4\lambda_{S}\lambda_{1}-\lambda_{S1}^{2})h_{1}^{4}+2\lambda_{S1}|\lambda_{S12}|\rho
h_{1}^{3}h_{2}$
$\displaystyle\quad+(4\lambda_{S}\lambda_{3}+4\lambda_{S}\lambda_{4}\rho^{2}-|\lambda_{S12}|^{2}\rho^{2}-2\lambda_{S1}\lambda_{S2})h_{1}^{2}h_{2}^{2}$
$\displaystyle\quad+2\lambda_{S2}|\lambda_{S12}|\rho
h_{1}h_{2}^{3}+(4\lambda_{S}\lambda_{2}-\lambda_{S2}^{2})h_{2}^{4}$
$\displaystyle=\lambda_{40}^{{}^{\prime}}h_{1}^{4}+\lambda_{31}^{{}^{\prime}}h_{1}^{3}h_{2}+\lambda_{22}^{{}^{\prime}}h_{1}^{2}h_{2}^{2}+\lambda_{13}^{{}^{\prime}}h_{1}h_{2}^{3}+\lambda_{04}^{{}^{\prime}}h_{2}^{4}.$

From the above expansion, let

$v_{1111}^{{}^{\prime}}=\lambda_{40}^{{}^{\prime}},\
v_{2222}^{{}^{\prime}}=\lambda_{04}^{{}^{\prime}},\
v_{1112}^{{}^{\prime}}=\displaystyle\frac{1}{4}\lambda_{31}^{{}^{\prime}},\
v_{1122}^{{}^{\prime}}=\frac{1}{6}\lambda_{22}^{{}^{\prime}},\
v_{1222}^{{}^{\prime}}=\frac{1}{4}\lambda_{13}^{{}^{\prime}}$

then we can regard $\widetilde{V}=(v^{{}^{\prime}}_{ijkl})$ as a 4th order
2-dimensional symmetric tensor, from the theorem 3.3, the expected conclusions
can be easily yield. ∎

In summary, we obtain an analytical necessary and sufficient condition of
copositivity for a special 4th order 3 dimensional symmetric tensor defined by
vacuum stability for $\mathbb{Z}_{3}$ scalar dark matter.

######  Theorem 3.6.

A tensor $\mathcal{V}=(v_{ijkl})$ given by (3.2) is copositive if and only if
$\lambda_{S}>0$ and

(1) $\lambda_{S1}\geq 0,$ $\lambda_{S2}\geq 0,$
$2\sqrt{\lambda_{S1}\lambda_{S2}}\geq|\lambda_{S12}|\rho$, $\lambda_{1}\geq
0,$ $\lambda_{2}\geq 0,$
$\lambda_{3}+\lambda_{4}\rho^{2}+2\sqrt{\lambda_{1}\lambda_{2}}\geq 0$.

(2) $\lambda_{S1}<0,$ $\lambda_{S2}<0$, $\lambda_{40}^{{}^{\prime}}>0,$
$\lambda_{04}^{{}^{\prime}}>0$ and

①$\Delta^{{}^{\prime}}\leq 0$,
$\lambda_{13}^{{}^{\prime}}\sqrt{\lambda_{40}^{{}^{\prime}}}+\lambda_{31}^{{}^{\prime}}\sqrt{\lambda_{04}^{{}^{\prime}}}>0$,
or

②$\lambda_{13}^{{}^{\prime}}\geq 0$, $\lambda_{31}^{{}^{\prime}}\geq 0$,
$\lambda_{22}^{{}^{\prime}}+2\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}\geq
0$, or

③$\Delta^{{}^{\prime}}\geq 0$,
$|\lambda_{31}^{{}^{\prime}}\sqrt{\lambda_{04}^{{}^{\prime}}}-\lambda_{13}^{{}^{\prime}}\sqrt{\lambda_{40}^{{}^{\prime}}}|\leq
4\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{22}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}+2\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}}$,

(i)$-2\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}\leq\lambda_{22}^{{}^{\prime}}\leq
6\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}$;

(ii)$\lambda_{22}^{{}^{\prime}}>6\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}$,

$\lambda_{31}^{{}^{\prime}}\sqrt{\lambda_{04}^{{}^{\prime}}}+\lambda_{13}^{{}^{\prime}}\sqrt{\lambda_{40}^{{}^{\prime}}}\geq-4\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{22}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}-2\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}\sqrt{\lambda_{40}^{{}^{\prime}}\lambda_{04}^{{}^{\prime}}}}$.

##  4 Conclusion

In this paper, with the help of Qi-Song-Zhang’s result, we present a simple
necessary and sufficient condition to test the copositivity of 4th order and
2-dimensional, and obtain an analytical necessary and sufficient condition of
copositivity for a special 4th order 3-dimensional symmetric tensor defined by
vacuum stability for $\mathbb{Z}_{3}$ scalar dark matter.

## References

  * [1] Balaji, R., Palpandi, K.: Positive definite and Gram tensor complementarity problems. Optim. Lett., 12(3), 639-648(2018)
  * [2] Bai, X.L., Huang, Z.H., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl., 170(1), 72-84(2016)
  * [3] Che, M., Qi, L., Wei, Y.: Stochastic $\mathcal{R}_{0}$ tensors to stochastic tensor complementarity problems, Optim. Lett., 13(2), 261-279(2019)
  * [4] Che, M.L., Qi, L.,Wei, Y.M.: Positive-definite tensors to nonlinear complementarity problems, J. Optim. Theory Appl., 168(2), 475-487(2016)
  * [5] Chen, H., Huang, Z., Qi, L.: Copositivity Detection of Tensors: Theory and Algorithm, J. Optim. Theory Appl., 174(3), 746-761(2017)
  * [6] Chen, H., Huang, Z.H., Qi, L.: Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69(1), 133-158(2018)
  * [7] Chen, H., Wang, Y.: High-order copositive tensors and its applications, J. Appl. Anal. Compu., 8(6), 1863-1885(2018)
  * [8] Chen, H., Qi, L., Song, Y.: Column suficient tensors and tensor complementarity problems, Front. Math. China, 13(2), 255-276(2018)
  * [9] Ding, W., Luo, Z., Qi, L.: $\mathcal{P}$-tensors,$\mathcal{P}_{0}$-tensors,and their applications. Linear Algebra Appl., 555, 336-354(2018)
  * [10] Gowda, M.S.: Polynomial complementarity problems. Pac. J. Optim. 13(2), 227-241(2017)
  * [11] Guo, Y.: A necessary and sufficient condition for the positive definite problem of a binary quartic form, arXiv:2009.01033[math.AG] 2 Sep 2020
  * [12] Hadeler, K.P.: On copositive matrices. Linear Algebra Appl., 49, 79-89(1983)
  * [13] G. Ulrich and L.T. Watson, Positivity conditions for quartic polynomials, SIAM J. Sci. Comput. 15, 528-544(1994)
  * [14] Huang, H., Qi, L.: Formulating an n-person noncooperative game as a tensor complementarity problem, Comput. Optim. Appl., 66, 557-576(2017)
  * [15] Kannike, K.: Vacuum stability of a general scalar potential of a few fields, Eur. Phys. J. C, 76, 324(2016)
  * [16] Kannike, K.: Erratum to: Vacuum stability of a general scalar potential of a few fields, Eur. Phys. J. C, 78, 355(2018)
  * [17] Kannike, K.: Vacuum stability conditions from copositivity criteria, Eur. Phys. J. C, 72, 2093(2012)
  * [18] Li, L., Zhang, X., Huang, Z., Qi, L.: Test of copositive tensors, J. Industrial Manag. Optim. 15(2), 881-891(2018)
  * [19] Liu, J., Song, Y.: Copositivity for 3rd order symmetric tensors and applications, arXiv: 1911.10284 ChinaXiv: 201911.00094 23 November 2019
  * [20] Luo, Z., Qi, L., Xiu, N.: The sparsest solutions to Z-tensor complementarity problems. Optim. Lett., 11(3), 471-482(2017)
  * [21] Nie, J., Yang, Z., Zhang, X.: A Complete Semidefinte algorithm for Detecting Copositive Matrices and Tensors, SIAM J. Optim.,28(4), 2902-2921(2018)
  * [22] Nadler, E.: Nonnegativity of bivariate quadratic functions on a triangle, Comput. Aided Geom.D., 9(3), 195-205(1992)
  * [23] Qi, L.: Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40, 1302-1324(2014)
  * [24] Qi, L.: Symmetric Nonnegative Tensors and Copositive Tensors, Linear Algebra Appl., 439,228-238(2013)
  * [25] Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications, Springer Singapore, (2018)
  * [26] Qi, L., Song, Y., Zhang, X.: Positivity Conditions for Cubic, Quartic and Quintic Polynomials, arXiv:2008.10922
  * [27] Qi, L., Luo, Z.: Tensor Analysis: Spectral Theory and Special Tensors. SLAM, Philadelphia, (2017)
  * [28] Song, Y., Qi, L.: Necessary and sufficient conditions of copositive tensors, Linear and Multilinear Algebra, 63(1), 120-131(2015)
  * [29] Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors, Ann. Appl. Math., 33(3), 308-323(2017)
  * [30] Song, Y., Qi, L.: Properties of some classes of structured tensors, J. Optim. Theory Appl., 165(3), 854-873(2015)
  * [31] Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors, J. Optim. Theory Appl., 169, 1069-1078(2016)
  * [32] Song, Y., Qi, L.: Analytical expressions of copositivity for 4th order symmetric tensors and applications, Analysis and Applications, DOI: 10.1142/S0219530520500049
  * [33] Song, Y., Qi, L.: Eigenvalue analysis of constrained minimization problem for homogeneous polynomial, J. Glob. Optim., 64(3), 563-575(2016)
  * [34] Song, Y., Yu, G.: Properties of solution set of tensor complementarity problem, J. Optim. Theory Appl., 170, 85-96(2016)
  * [35] Song, Y., Qi, L.: A necessary and sufficient condition of positive definiteness for 4th order symmetric tensors defined in particle physics. arXiv: 2011.11262
  * [36] Song, Y., Qi, L.: Strictly semi-positive tensors and the boundedness of tensor complementarity problems, Optim. Lett., 11, 1407-1426(2017)
  * [37] Song, Y.: Positive definiteness for 4th order symmetric tensors and applications. Anal. Math. Phys., 11, 10(2021)
  * [38] Song, Y., Mei, W.: Structural Properties of Tensors and Complementarity Problems. J. Optim. Theory Appl., 176(2), 289-305(2018)
  * [39] Wang, X., Chen, H. Wang, Y.: Solution structures of tensor complementarity problem. Front. Math. China, 13(4), 935-945(2018)
  * [40] Wang, Y., Huang, Z.H., Bai, X.L.: Exceptionally regular tensors and tensor complementarity problems. Optim. Method. Softw., 31(4), 815-828(2016)
  * [41] Wang, J., Hu, S., Huang, Z.H.: Solution sets of quadratic complementarity problems. J. Optim. Theory Appl., 176(1), 120-136(2018)