Multi-site disease confrontation detection method based on function connection
By adopting a multi-site anti-disease detection method based on functional connection in autism diagnosis, using autoencoder and one-dimensional convolutional neural network to extract the features of the functional connection matrix, and designing an adversarial learning network architecture, the problem of failure to fully consider statistical confusion effects and site information interference in the existing technology is solved, and a higher accuracy of autism classification is achieved.
Patent Information
- Application Number
- CN202411710874.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-11-27
- Publication Date
- 2025-03-11
- Estimated Expiration
- Not applicable · inactive patent
AI Technical Summary
The existing autism diagnosis method fails to fully consider the confusion effect in statistics and site information interference, resulting in controversy in the repetition and accuracy of the diagnostic results.
A multi-site anti-disease detection method based on functional connection is adopted, and the features of the functional connection matrix are extracted through an autoencoder and one-dimensional convolutional neural network, feature selection is performed by combining site features and scale information, and an adversarial learning network architecture is designed to alternately train site feature regression and disease classification tasks to balance the losses of the two to improve disease detection performance.
It effectively improves the classification accuracy of autism, reduces the impact of site confounding factors on model feature extraction, and enables the model to achieve a Nash equilibrium between removing site factors and disease classification.
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Figure CN119600357A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of medical image processing and analysis, and specifically relates to a multi-site disease detection method based on functional connection. Background Art
[0002] Autism spectrum disorder (ASD) is a neurodevelopmental disorder that affects an individual's language, social skills, and behavioral patterns. This disorder usually appears in early childhood and may last a lifetime, with significant impacts on patients, their families, and society. Previous autism diagnostic methods were guided by symptom-based clinical criteria, which required a large number of behavioral assessments and high professional knowledge of doctors. In addition, the results of the diagnosis were affected by the subjectivity of the doctor, which may lead to misdiagnosis and delayed diagnosis. In recent years, the diagnosis rate of autism has continued to rise, attracting widespread attention. Medical imaging technologies, such as magnetic resonance imaging (MRI), play an important role in the study of brain diseases such as autism.
[0003] MRI can be divided into sMRI (structural magnetic resonance imaging) and fMRI (functional magnetic resonance imaging) according to its imaging purpose and technical characteristics. Structural magnetic resonance imaging, with its high clarity and resolution, can provide three-dimensional images of brain structure, helping doctors to conduct quantitative analysis of specific brain areas. Functional magnetic resonance imaging reflects the activity state of the brain by monitoring changes in blood oxygen levels, and is a powerful tool for studying neurological diseases such as autism. fMRI includes resting-state functional magnetic resonance imaging (rs-fMRI) and task-state functional magnetic resonance imaging (t-fMRI), the latter of which can provide brain activity data in different states. Among them, the use of functional connectivity data from resting-state functional magnetic resonance imaging to assist in the diagnosis of diseases has become the mainstream.
[0004] Since the amount of rs-fMRI data from a single site is usually small, the reproducibility and generalizability of the study may be controversial. Using multi-site data to form a large data set has become an effective way to solve such problems. The Autism Brain Imaging Data Exchange (ABIDE) program aggregates functional and structural brain imaging data collected by laboratories around the world. However, different sites usually have differences in acquisition equipment, acquisition parameters, and acquisition populations, which makes multi-site data sets inherently heterogeneous. The confounding effect caused by multiple sites will affect the relationship between input data and output variables, resulting in a correlation between site differences and biological predictions. This correlation will hinder the estimation of true biological changes, or infer non-biological differences as biological differences, causing the model to produce false or biased predictions.
[0005] New neuroimaging technologies provide a multi-angle perspective for studying the pathogenesis of autism, and the data obtained help to discover potential neural biomarkers, which are crucial for the early diagnosis and treatment of autism. In recent years, based on these neuroimaging data, deep learning technology has been used to process them, which has improved the accuracy and efficiency of diagnosis. However, existing methods do not fully consider factors such as statistical confounding effects and site information interference. Therefore, it is necessary to provide a new disease classification method to solve the above problems. Summary of the invention
[0006] In view of the technical problem that the above-mentioned existing methods do not fully consider factors such as the confusion effect in statistics and the interference of site information, the present invention provides a multi-site disease detection method based on functional connectivity to achieve more accurate classification detection of autism.
[0007] In order to solve the above technical problems, the technical solution adopted by the present invention is:
[0008] A multi-site disease detection method based on functional connectivity, comprising the following steps:
[0009] S1. Data acquisition and preprocessing: Use magnetic resonance imaging equipment to obtain brain rs-fMRI data of patients and normal subjects in multiple imaging centers, perform time correction, head motion correction, and normalization preprocessing, and finally calculate the functional connectivity matrix;
[0010] S2, site feature learning: extract features from the functional connectivity data of each site through an autoencoder, generate a low-dimensional feature vector, and obtain the site feature vector through average pooling; at the same time, perform feature selection in combination with scale information;
[0011] S3, representation learning: use one-dimensional convolutional neural network 1D-CNN to extract features from the functional connection matrix, ensure that the convolution operation adapts to the relationship between brain regions, and output the features required for classification;
[0012] S4, adversarial training: design an adversarial learning network architecture, combine site feature regression and disease classification tasks for alternating training, and balance the two by optimizing the objective function to improve disease detection performance;
[0013] S5. Disease detection: Divide the dataset into training set, test set and validation set, and use ten-fold cross validation to evaluate the model performance to achieve disease detection.
[0014] The data standard in S1 is: having complete scale information.
[0015] The site feature learning in S2 quantifies the heterogeneity between different sites to generate a feature vector; the autoencoder AE and site average pooling are used to extract the image features of the functional connection; the autoencoder includes an encoder and a decoder, and generates a low-dimensional feature vector by minimizing the reconstruction error training;
[0016] To extract the subject-level feature vector, after deleting redundant information, the upper triangular part of the functional connection matrix is flattened and input into AE for training to obtain the subject-level feature vector; then, the mean vector is calculated by site average pooling to form the site feature vector; the encoder encodes the input data into a low-dimensional representation, the formula is: h enc =φ enc (x) = f(W enc x+b enc ), where f is the activation function, W enc and b enc They are the weight and bias of the encoder respectively; the decoder reconstructs the output of the encoder back to the original input data, the formula is: x′=φ dec (h enc )=W dec h enc +b dec ;W dec and b dec are the weights and biases of the decoder respectively; in addition, potential site differences are extracted from the scale information of the subjects; due to the small dimension of the scale, direct splicing may not be effective, so feature selection is performed based on the cosine similarity between the site feature vector and the scale information, and the mean and variance vectors are calculated and standardized; finally, the single-dimensional feature vector with high cosine similarity is selected as the site feature, and the scale information includes gender, age, full-scale IQ, verbal IQ and performance IQ.
[0017] The input of the representation learning module in S3 is the functional connectivity matrix, which represents the brain functional network. Due to its non-Euclidean spatial characteristics, the traditional CNN is not effective in feature extraction. A one-dimensional convolutional neural network 1D-CNN is designed as a representation learning module, which includes two convolutional layers and one fully connected layer.
[0018] The first layer of the representation learning module uses 64 1×200 horizontal convolution kernels, where 200 is the number of brain regions, 64 is the number of channels, and each row represents the correlation between brain regions; this layer generates 64 feature maps with a shape of 64×200×1; the second layer uses 128 200×1 vertical convolution kernels to extract whole-brain features from brain region features, with an output dimension of 128; no pooling layer is used to avoid redundant features and loss of key features; the convolution operation formula is: in is the output of the current layer, is the output of the previous layer, is the parameter of the current layer, × is the dot product operation of the corresponding receptive field, is the bias of the current layer; the activation function uses tanh to adapt to the value range of the connection matrix (-1 to 1); the classification layer uses Softmax to calculate the category probability, and uses the DropOut layer and L2 regularization to prevent overfitting, and the InstanceNorm layer is used to normalize each channel.
[0019] The objective function of the adversarial learning network architecture in S4 is composed of the classification and site feature regression loss functions. The purpose is to reduce the classification loss while increasing the site feature regression loss. Therefore, the site feature regression in the objective function is called site feature regression adversarial; X = {X1, ..., X N} represents the functional connection of N subjects; Y = {Y1, ..., Y N} represents the labels of N subjects; C = {C1, ..., C N} represents the site characteristics of N subjects; θ E is the parameter that characterizes the learning module; θ P are the classifier parameters; θ R is the regression model parameter; the site feature regression task loss function is the mean square error, and the formula is: The loss function for the classification task is cross entropy loss, and the formula is:
[0020] The method of combining site feature regression and disease classification tasks for alternating training in S4 is as follows: the model adopts an alternating iterative training method, and in each epoch, the regression task loss L is first optimized. R , train the regression model parameters; then optimize the objective function loss L t , training representation learning module and classifier parameters; in order to achieve confrontation between the two trainings, the objective function is to minimize the classification loss L P At the same time, we also need to maximize the regression task loss L R ; Therefore, we choose to use the regression task loss function as the denominator of a fraction and then add the entire fraction to the classification task loss function. In this way, when the optimizer reduces the objective function, the classification task loss L P It also reduces the regression task loss L R Increase, as shown below; α is a hyperparameter used to balance the loss optimization of the two tasks and prevent L R Excessive increase leads to excessive restrictions on the representation learning module, and the optimal classification performance cannot be achieved; the final objective function is expressed as follows:
[0021] Compared with the prior art, the present invention has the following beneficial effects:
[0022] Considering the significant differences between data collected from different sites, the present invention provides a multi-site adversarial architecture autism classification detection method, which extracts non-redundant common features within the site through AE, site feature average pooling and feature selection, and then uses the newly proposed adversarial loss function to remove the influence of site confounding factors on feature extraction of the representation learning module during training, so that the model reaches Nash equilibrium between removal and disease classification. The data used in the present invention does not rely too much on clinical information and uses multi-site data, which can effectively improve the accuracy of autism classification and play an important role in disease diagnosis. BRIEF DESCRIPTION OF THE DRAWINGS
[0023] In order to more clearly illustrate the implementation methods of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for the implementation methods or the description of the prior art. Obviously, the drawings in the following description are only exemplary, and for ordinary technicians in this field, other implementation drawings can be derived from the provided drawings without creative work.
[0024] The structures, proportions, sizes, etc. illustrated in this specification are only used to match the contents disclosed in the specification so as to facilitate understanding and reading by persons familiar with the technology. They are not used to limit the conditions under which the present invention can be implemented, and therefore have no substantial technical significance. Any structural modification, change in proportion or adjustment of size shall still fall within the scope of the technical contents disclosed in the present invention without affecting the effects and purposes that can be achieved by the present invention.
[0025] Figure 1 A schematic flow chart of a multi-site disease detection method based on functional connectivity according to an embodiment of the present invention;
[0026] Figure 2 Schematic diagram of the process of preprocessing resting state functional magnetic resonance data in an embodiment of the present invention;
[0027] Figure 3 A schematic diagram of the principle of site feature learning in an embodiment of the present invention;
[0028] Figure 4 Schematic diagram of the principle of representation learning in an embodiment of the present invention;
[0029] Figure 5 Schematic diagram of the adversarial training process in an embodiment of the present invention. DETAILED DESCRIPTION
[0030] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. These descriptions are only to further illustrate the features and advantages of the present invention, rather than to limit the claims of the present invention. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of this application.
[0031] The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings and examples. The following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.
[0032] The embodiment of the present invention relates to a multi-site disease detection method based on functional connectivity, such as Figure 1 As shown, the following steps are included:
[0033] Step S1, data collection and preprocessing: Figure 2 As shown, MRI equipment was used to collect brain rs-fMRI data of patients and normal subjects and the corresponding scale information in multiple imaging centers; the collected data were preprocessed including time correction, head motion correction, normalization, regression linear drift, bandpass filtering (0.01-0.1Hz), and standard spatial registration; the Craddock 200 brain atlas template was used to divide the brain region of interest (ROI), and the average BOLD signal time series of each ROI was calculated. The Pearson correlation coefficient was used to evaluate the functional connectivity between each pair of ROI average time series, and finally a 200*200 functional connectivity matrix of each subject was obtained based on the CC200 brain atlas;
[0034] Step S2, site feature learning: An autoencoder (AE) is used to extract implicit site information features from the functional connectivity data of all subjects belonging to the same site. All features are integrated into representative features of the site through average pooling. Finally, the abstract site confounding factors are instantiated as features in vector form;
[0035] Step S3, representation learning: abstract features containing physiological meanings are extracted from the functional connectivity data of each subject through CNN convolutional layers and MLP layers;
[0036] Step S4, adversarial training: Inspired by the generative adversarial network (GAN), a network architecture for adversarial learning is designed. During the training process, two tasks, site feature regression and disease classification, are performed simultaneously, and the training process is constrained by a newly designed loss function. The two tasks are trained alternately and eventually reach a Nash equilibrium, thereby enhancing disease detection performance;
[0037] Step S5, disease detection: divide the data set into training set, test set and validation set. Use ten-fold cross validation to complete disease detection for all subjects, and evaluate model performance by classification accuracy.
[0038] Specifically, Figure 3 As shown, in step S2, site feature learning aims to quantify the heterogeneity between different sites to form a feature vector for representing site differences. The present invention uses an autoencoder (AE) and site average pooling to extract image features of sites in functional connections. The autoencoder consists of an encoder and a decoder, and is trained by minimizing the reconstruction error to generate a low-dimensional feature vector.
[0039] To extract the subject-level feature vector, after removing the redundant information of the functional connectivity matrix, the upper triangular part is flattened and input into the AE training. After the training is completed, the encoder output obtained is the subject-level feature vector. Then, the mean vector of each site is calculated by site average pooling to form a site feature vector. The encoder encodes the input data into a low-dimensional representation. This step can be expressed by the formula:
[0040] h enc =φ enc (x) = f(W enc x+b enc )
[0041] Where f is the activation function, W enc and b enc are the weights and biases of the encoder respectively. The decoder reconstructs the output of the encoder back to the original input data, as shown below, where W dec and b dec are the weights and biases of the decoder respectively.
[0042] x′=φ dec (h enc )=W dec h enc +b dec
[0043] In addition, potential site differences can also be extracted from the test scale information. Since the scale information dimension is small, direct splicing may not effectively reflect the difference. Therefore, feature selection is performed based on the cosine similarity between the site feature vector and the scale information, and the mean and variance vectors are calculated and standardized. The cosine similarity calculation method is as follows, where A and B represent two vectors.
[0044]
[0045] Finally, the single-dimensional feature vector with high cosine similarity is selected as the final site feature. The scale information used in the present invention includes gender, age, full scale IQ, verbal IQ and performance IQ.
[0046] Specifically, Figure 4 As shown in Figure 1, in step S3, the input of the representation learning module is the functional connectivity matrix, which represents the brain functional network. Its data structure is different from that of the image: the former is a non-Euclidean space, while the latter is a Euclidean space. Therefore, the feature extraction effect of the traditional CNN on the brain functional network may not be good. Therefore, based on the topological structure of the brain functional network, a one-dimensional convolutional neural network (1D-CNN) is designed as a representation learning module. The one-dimensional convolution kernel is used to perform convolution operations on the functional connectivity matrix. It consists of two 1D-CNN convolutional layers and one fully connected layer.
[0047] The first layer uses a horizontal convolution kernel of 64@1*200, where 200 is the number of brain regions and 64 is the number of channels. Each row represents the correlation between a certain brain region and other brain regions, ensuring the relationship between the target brain region and its adjacent brain regions during convolution calculation. After the first layer of convolution, 64 feature maps with a shape of 64*200*1 are generated. The second layer uses a vertical convolution kernel of 128@200*1 to extract whole-brain features from brain region features, and the output dimension is 128. The pooling layer is not used in the model because the convolution operation is based on brain regions and is independent of each other, avoiding redundant features and reducing the loss of key features. The convolution operation is shown below, where is the output of the current layer, is the output of the previous layer, is the parameter of the current layer, × is the dot product operation of the corresponding receptive field, is the bias of the current layer.
[0048]
[0049] The activation function is tanh to adapt to the range of the functional connection matrix (-1 to 1). The classification layer uses the Softmax function to calculate the category probability, and uses the DropOut layer and L2 regularization to prevent overfitting. At the same time, the InstanceNorm layer is used to normalize the features of each channel.
[0050] Specifically, in step S4, the overall process of implementing the training is as follows: Figure 5As shown. Existing methods usually use a zero-sum game approach to train models, optimizing the adversarial task of removing confounding effects separately to minimize its loss value. However, when the target task is inconsistent with the adversarial task, minimizing the loss of the adversarial task may not necessarily optimize the target task, and may even have a negative impact. Studies have found that adversarial tasks that remove scanner information can also reduce prediction performance. Therefore, the method of the present invention does not train the adversarial task separately, but combines it with the target task into an objective function, and balances the training of the two by optimizing the loss of the objective function to improve the performance of the target task.
[0051] The objective function is a combination of the classification and site feature regression loss functions. The purpose is to reduce the classification loss while increasing the site feature regression loss. Therefore, the site feature regression in the objective function is called site feature regression adversarial. N} represents the functional connection of N subjects; Y = {Y1, ..., Y N} represents the labels of N subjects; C = {C1, ..., C N} represents the site characteristics of N subjects; θ E is the parameter that characterizes the learning module; θ P are the classifier parameters; θ R is the regression model parameter. The loss function of the site feature regression task is the mean square error, and the formula is:
[0052]
[0053] The loss function for the classification task is cross entropy loss, and the formula is:
[0054]
[0055] The model adopts an alternating iterative training method, and in each epoch, the regression task loss L is first optimized. R , train the regression model parameters; then optimize the objective function loss L t , training representation learning module and classifier parameters. In order to achieve confrontation between the two trainings, the objective function is to minimize the classification loss L P At the same time, we also need to maximize the regression task loss L R Therefore, we choose to use the regression task loss function as the denominator of a fraction and then add the entire fraction to the classification task loss function. In this way, when the optimizer reduces the objective function, the classification task loss L P It also reduces the regression task loss L R Increase, as shown below. α is a hyperparameter used to balance the loss optimization of the two tasks and prevent L R Excessive increase will lead to excessive restrictions on the representation learning module and fail to achieve the optimal classification performance.
[0056]
[0057] The present invention provides a new disease classification method that eliminates the influence of site factors. The features used do not rely too much on clinical information and use data from multiple sites, which can effectively improve the accuracy of autism classification. After ten-fold cross-validation, the disease classification model proposed by the present invention has an average accuracy of 75.6% on the ABIDE I dataset, proving that the disease classification model proposed by the present invention has relatively good classification ability.
[0058] Only the preferred embodiments of the present invention are described in detail above, but the present invention is not limited to the above embodiments. Various changes can be made within the knowledge scope of ordinary technicians in this field without departing from the purpose of the present invention, and various changes should be included in the protection scope of the present invention.
Claims
1. A multi-site disease detection method based on functional connectivity, characterized in that: The following steps are involved: S1. Data acquisition and preprocessing: Use magnetic resonance imaging equipment to obtain brain rs-fMRI data of patients and normal subjects in multiple imaging centers, perform time correction, head motion correction, and normalization preprocessing, and finally calculate the functional connectivity matrix; S2, site feature learning; The autoencoder is used to extract features from the functional connectivity data of each site, generate a low-dimensional feature vector, and obtain the site feature vector through average pooling; at the same time, feature selection is performed in combination with the scale information; S3, representation learning: use one-dimensional convolutional neural network 1D-CNN to extract features from the functional connection matrix, ensure that the convolution operation adapts to the relationship between brain regions, and output the features required for classification; S4, adversarial training: design an adversarial learning network architecture, combine site feature regression and disease classification tasks for alternating training, and balance the two by optimizing the objective function to improve disease detection performance; S5. Disease detection: Divide the dataset into training set, test set and validation set, and use ten-fold cross validation to evaluate the model performance to achieve disease detection.
2. According to the multi-site disease detection method based on functional connectivity according to claim 1, it is characterized in that: The data standard in S1 is: having complete scale information.
3. The multi-site disease detection method based on functional connectivity according to claim 1, characterized in that: The site feature learning in S2 quantifies the heterogeneity between different sites to generate feature vectors; the autoencoder AE and site average pooling are used to extract the image features of functional connections; The autoencoder consists of an encoder and a decoder, which are trained by minimizing the reconstruction error to generate a low-dimensional feature vector; To extract the subject-level feature vector, after removing redundant information, the upper triangular part of the functional connectivity matrix is flattened and input into AE for training to obtain the subject-level feature vector; Then, the mean vector is calculated by site average pooling to form a site feature vector; the encoder encodes the input data into a low-dimensional representation, the formula is: enc =φ enc (x) = f(W enc x+b enc );in f is the activation function, W enc and b enc are the weights and biases of the encoder respectively; The decoder reconstructs the encoder output back to the original input data, the formula is: x′=φ dec (h enc )=W dec h enc +b dec ;W dec and b dec are the weights and biases of the decoder respectively; in addition, potential site differences are extracted from the scale information of the subjects; due to the small dimension of the scale, direct splicing may not be effective, so feature selection is performed based on the cosine similarity between the site feature vector and the scale information, and the mean and variance vectors are calculated and standardized; finally, the single-dimensional feature vector with high cosine similarity is selected as the site feature, and the scale information includes gender, age, full-scale IQ, verbal IQ and performance IQ.
4. The multi-site disease detection method based on functional connectivity according to claim 1, characterized in that: The input of the representation learning module in S3 is the functional connectivity matrix, which represents the brain functional network. Due to its non-Euclidean spatial characteristics, the traditional CNN is not effective in feature extraction. A one-dimensional convolutional neural network 1D-CNN is designed as a representation learning module, which includes two convolutional layers and one fully connected layer.
5. The multi-site disease detection method based on functional connectivity according to claim 4, characterized in that: The first layer of the representation learning module uses 64 1×200 horizontal convolution kernels, where 200 is the number of brain regions, 64 is the number of channels, and each row represents the correlation between brain regions; This layer produces 64 feature maps with a shape of 64×200×1; The second layer uses 128 200×1 vertical convolution kernels to extract the whole brain features from the brain region features, and the output dimension is 128; no pooling layer is used to avoid the loss of redundant features and key features; the convolution operation formula is: in is the output of the current layer, is the output of the previous layer, is the parameter of the current layer, × is the dot product operation of the corresponding receptive field, is the bias of the current layer; the activation function uses tanh to adapt to the value range of the connection matrix (-1 to 1); the classification layer uses Softmax to calculate the category probability, and uses the DropOut layer and L2 regularization to prevent overfitting, and the InstanceNorm layer is used to normalize each channel.
6. The multi-site disease detection method based on functional connectivity according to claim 1, characterized in that: The objective function of the adversarial learning network architecture in S4 is composed of the classification and site feature regression loss functions. The purpose is to reduce the classification loss while increasing the site feature regression loss. Therefore, the site feature regression in the objective function is called site feature regression adversarial; X = {X1, ..., XN} represents the functional connection of N subjects; Y = {Y1, ..., YN} represents the labels of N subjects; C = {C1, ..., C N } represents the site characteristics of N subjects; θ E is the parameter that characterizes the learning module; θ P are the classifier parameters; θ R is the regression model parameter; the site feature regression task loss function is the mean square error, and the formula is: The loss function for the classification task is cross entropy loss, and the formula is:
7. The multi-site disease detection method based on functional connectivity according to claim 1, characterized in that: The method of combining site feature regression and disease classification tasks for alternating training in S4 is as follows: the model adopts an alternating iterative training method, and in each epoch, the regression task loss L is first optimized. R , train the regression model parameters; then optimize the objective function loss L t , training representation learning module and classifier parameters; in order to achieve confrontation between the two trainings, the objective function is to minimize the classification loss L P At the same time, we also need to maximize the regression task loss L R ; Therefore, we choose to use the regression task loss function as the denominator of a fraction and then add the entire fraction to the classification task loss function. In this way, when the optimizer reduces the objective function, the classification task loss L P It also reduces the regression task loss L R Increase, as shown below; α is a hyperparameter used to balance the loss optimization of the two tasks and prevent L R Excessive increase leads to excessive restrictions on the representation learning module, and the optimal classification performance cannot be achieved; the final objective function is expressed as follows:
Citation Information
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