A ReduNet white-box algorithm based on normalization in a spherical coordinate system

Abstract

The application discloses a kind of normalization-based ReduNet white-box algorithm under spherical coordinate system, belong to ReduNet white-box network field, this method includes obtaining the input data of sample size m under n-dimensional space, and input data is converted from Cartesian coordinate system to spherical coordinate system, obtain the input data under spherical coordinate system;Radial coordinate is 1, define the input data under spherical coordinate system on unit sphere, obtain the normalization data under spherical coordinate system;The normalization data under spherical coordinate system is input as initial feature ReduNet white-box network.The application solves the problem that the feature updated by each layer needs to be normalized by projecting onto the unit sphere when the existing normalization method is applied in ReduNet white-box network.

Description

Technical Field

[0001] This invention belongs to the field of ReduNet white-box networks, and particularly relates to a normalized ReduNet white-box algorithm based on spherical coordinates. Background Technology

[0002] In recent years, deep neural networks have achieved tremendous success in many fields, such as image recognition, speech processing, natural language processing, and Go. The practice of deep networks has captured the imagination with its experience in learning useful representations from such large-scale real-world data. To a large extent, the rise of deep neural networks is attributed to significant technological advancements in scalable computing platforms, and equally importantly, to the abundance of real or simulated data for network training and evaluation. However, the practice of deep networks has been shrouded in uninterpretability from the outset, largely because deep network architectures and their components are typically designed based on years of trial and error, trained from random initializations via backpropagation, and deployed as “black boxes.” Many popular techniques and methods for designing and training deep networks have been developed through heuristics and empirical approaches rather than rigorous mathematical principles, modeling, and analysis. To demystify deep networks, a recent body of research on the mathematics of deep learning has greatly improved our understanding of key aspects of deep models, including overfitting and overparameterization, and optimization. However, these efforts are often based on simplified models, such as networks with only a few layers; or rely on simplified assumptions, such as ultrawide deep networks; or only explain or prove one of the network's components or features, such as random deactivation methods.

[0003] On the other hand, the dominant approach in the above practices remains the black-box structure of backpropagation through trial and error. Therefore, the white-box neural network ReduNet aims to explain modern deep convolutional networks from the principles of data compression and discriminative representation. From this goal, a basic iterative gradient ascent scheme for optimizing rate reduction is derived, known as the Maximum Rate Descent Principle (MCR2).

[0004] Optimizing this objective will naturally generate a result such as Figure 1 The multi-layered "white-box" deep network shown is called ReduNet, which possesses the common characteristics of modern deep networks. Its deep architecture, linear and nonlinear operators, and even the network parameters are explicitly constructed layer by layer through forward propagation. All parts have precise optimization, statistical, and geometric interpretations, and the effectiveness of the rate reduction objective and the white-box network has been verified in existing technical solutions.

[0005] By observing the structure of the network, it can be seen that each layer of the network ultimately needs to pass through... This layer, that is, the updated features of each layer always need to be projected onto the unit sphere S. n-1 Normalization can be understood as performing a unit norm normalization operation on Z of each layer after each update of new features, so that the dimensions of each layer are consistent. This step is the normalization method for feature scaling during the training of deep neural networks.

[0006] However, the large variance of neurons in neural networks makes the model highly sensitive to changes in the input distribution, leading to poor generalization. Furthermore, as information moves forward through layers, the variance can be amplified, especially in deep networks where large variance exacerbates the shift in internal covariates. That is, as the parameters of the previous layer change, the distribution of each layer changes during training, and the shift in internal covariates slows down training because each layer needs to constantly adapt to new feature distributions. Traditionally, neural networks use dot products to calculate the pre-activation of neurons. The result of a dot product is unbounded, meaning it can be any value in the entire real space, increasing the risk of large variance. Using small weights can alleviate these problems; weight decay and maximum normalization are methods that attempt to reduce weights. However, when using weight decay or maximum normalization, the objective (or objective direction) of the original optimization problem changes, and they also introduce additional hyperparameters.

[0007] Batch normalization normalizes pre-activations using statistics computed from mini-batch training examples. The normalized values ​​are rescaled and re-shifted using additional parameters. Because batch normalization uses statistics from mini-batch examples, its effectiveness depends on the batch size. To overcome this, normalization propagation uses data-independent mean parameter estimates, while layer normalization computes the mean and standard deviation of the same layer in a single training case. Weight normalization reparameterizes the input by partitioning it into a weight vector. It decouples the length of the weight vector from its direction, thus partially constraining the result of the dot product, but without considering the length of the input vector. All of these methods introduce additional parameters that need to be learned, making the model more complex.

[0008] The normalization operation in the ReduNet white-box network is essentially the batch normalization and layer normalization methods mentioned above in neural networks. Since the updated features of each layer always need to be normalized by projecting onto a unit sphere, this raises a new perspective: Is there a good normalization method that can accelerate the convergence speed of the neural network while removing the unitization operation of each layer? Or is it possible to create a new network that can be trained using gradient ascent in white-box networks, but does not depend on any statistics of large or small batches in the above normalization methods?

[0009] Based on this idea, one existing technical solution proposes using cosine similarity or center cosine similarity, represented only by angles, to replace the dot product in neural networks, a technique known as cosine normalization. Another existing technical solution proposes a novel hyperspherical convolutional learning framework and provides angle representations on the hypersphere. The sphere specification introduced in this hyperspherical convolutional network can serve as a novel normalization method. Yet another existing technical solution proposes a spherical layer to represent all angle similarity information. Unlike other projection-based methods, this method ensures that representation learning always occurs on the hypersphere, completely eliminating the interference of normalization operations during training without losing any information. Furthermore, this method can be applied to any network by replacing existing layers, validating the broad applicability and scalability of the spherical layer.

[0010] Converting data representations from Cartesian coordinates to spherical coordinates in traditional deep learning methods expands the application scope of deep learning across different data representations. However, traditional normalization methods face challenges in interpreting models in white-box neural networks due to potential numerical instability and gradient vanishing issues when processing data in spherical coordinates. This raises the question of whether introducing novel normalization methods in spherical coordinates can enhance interpretability, improve test accuracy, and accelerate convergence in white-box networks. Therefore, researching normalization methods based on spherical coordinates and their applications in white-box neural networks is highly significant.

[0011] Existing batch normalization methods must calculate the mean and variance to normalize the previous outputs within a batch. This statistical estimation is relatively accurate when the batch size is large, but the accuracy decreases continuously as the batch size decreases. Because each layer requires normalization of a mini-batch of data, and although batch normalization methods are not computationally intensive and reduce the total number of iterations required for convergence, the time required for each iteration increases significantly, and this increase further with increasing batch size.

[0012] Layer normalization requires calculating the mean and variance of the activations for each layer, which increases computational overhead. While these calculations are negligible in most cases, they can impact overall performance in environments with limited computational resources.

[0013] While layer normalization can improve training stability, it may not significantly speed up training as much as batch normalization. Batch normalization uses statistical information from small batches of data for normalization and typically converges faster. Layer normalization, on the other hand, requires normalization for each individual sample, which can lead to relatively slower convergence. Summary of the Invention

[0014] To address the aforementioned shortcomings in existing technologies, this invention provides a ReduNet white-box algorithm based on normalization in spherical coordinates. This solves the problem that existing normalization methods, when applied to ReduNet white-box networks, always require normalization by projecting the updated features onto a unit sphere.

[0015] To achieve the aforementioned objectives, the present invention employs the following technical solution: a normalized ReduNet white-box algorithm based on spherical coordinates, comprising the following steps:

[0016] S1. Obtain input data of sample size m in n-dimensional space, and transform the input data from Cartesian coordinate system to spherical coordinate system to obtain input data in spherical coordinate system;

[0017] S2. Set the radial coordinate to 1, define the input data in the spherical coordinate system on the unit sphere, and obtain the normalized data in the spherical coordinate system.

[0018] S3. Normalized data in spherical coordinates is used as the initial feature input to the ReduNet white-box network.

[0019] Furthermore, the expression for the input data in the spherical coordinate system in step S1 is:

[0020]

[0021] Where X represents the input data in spherical coordinates; x1 represents the 1D component of the input data in Cartesian coordinates; x2 represents the 2D component of the input data in Cartesian coordinates; x n-1 The input data consists of n-1 dimensional components in Cartesian coordinates; x n θ represents the n-dimensional components of the input data in Cartesian coordinates; θ1 represents the angular coordinates corresponding to the 1-dimensional components of the input data; θ n-2 θ represents the angular coordinates corresponding to the n-2 dimensional components of the input data. n-1 r represents the angular coordinates corresponding to the n-1 dimensional components of the input data; r is the radial coordinate.

[0022] Furthermore, the expression for the normalized data in the spherical coordinate system in step S2 is:

[0023]

[0024] Where θ represents the normalized data in the spherical coordinate system; θ1 represents the angular coordinates corresponding to the 1-dimensional components of the input data; θ2 represents the angular coordinates corresponding to the 2-dimensional components of the input data; θ3 represents the angular coordinates corresponding to the 3-dimensional components of the input data; θ n-2 θ represents the angular coordinates corresponding to the n-2 dimensional components of the input data. n-1x1 represents the angular coordinates corresponding to the (n-1)-dimensional components of the input data; x2 represents the 2-dimensional component of the input data in Cartesian coordinates; x3 represents the 3-dimensional component of the input data in Cartesian coordinates; x n-1 The input data consists of n-1 dimensional components in Cartesian coordinates.

[0025] Furthermore, in step S3, the gradient for solving the maximum bitrate problem using the gradient ascent method in the ReduNet white-box network is:

[0026]

[0027]

[0028] Where R is the coding rate; The sign for the partial derivative is θ; θ represents the normalized data in spherical coordinates; θ 1 The normalized data is in the spherical coordinate system of the first sample; θ 2 The normalized data is in the spherical coordinate system of the second sample; θ m Z represents the normalized data of the m-th sample in spherical coordinates; m is the total number of samples; Z 1 Z represents the features of the first sample. 2 Features of the second sample; Z m G represents the features of the m-th sample; 1 G is the transpose of the derivative matrix of the first sample; 2 G is the transpose of the derivative matrix of the second sample; m G is the transpose of the derivative matrix of the m-th sample; T is the transpose; G i J is the transpose of the derivative matrix of the i-th sample; i Z is the derivative matrix for the i-th sample; i θ represents the features of the i-th sample; i θ1 represents the normalized data in the spherical coordinate system of the i-th sample; θ2 represents the angular coordinates corresponding to the 1-dimensional components of the input data; θ3 represents the angular coordinates corresponding to the 3-dimensional components of the input data; θ4 represents the normalized data in the spherical coordinate system of the i-th sample. p θ represents the angular coordinates corresponding to the p-dimensional components of the input data. n-1 The input data consists of n-1 dimensional components with angular coordinates; n is the spatial dimension; and i is the sample number.

[0029] Furthermore, the gradient ascent formula for the ReduNet white-box network in step S3 is:

[0030]

[0031] Where, θ l+1 θ represents the angular coordinates of the (l+1)th layer of the network.l Let be the angular coordinates of the l-th layer network; η be the learning rate; and R be the encoding rate. is the sign of the partial derivative; θ represents the normalized data in spherical coordinates. For the case of the l-th layer network; Z 1 Z represents the features of the first sample. 2 Features of the second sample; Z m G represents the features of the m-th sample; 1 G is the transpose of the derivative matrix of the first sample; 2 G is the transpose of the derivative matrix of the second sample; m It is the transpose of the derivative matrix of the m-th sample.

[0032] The beneficial effects of this invention are as follows: Based on a novel normalization method in spherical coordinates, this invention fully utilizes the distribution characteristics of spherical coordinate data, compressing the data distribution onto a unit sphere to improve the numerical stability and convergence speed of the model. Furthermore, embedding the newly designed normalization method in spherical coordinates into a white-box neural network improves the network's interpretability, test accuracy, and convergence speed under different data representations. Attached Figure Description

[0033] Figure 1 This is a schematic diagram of the ReduNet white-box network in the background technology of this invention.

[0034] Figure 2 This is a flowchart of the method of the present invention.

[0035] Figure 3 This is a graph showing the loss function of the spherical coordinate normalized network in this embodiment of the invention as iterates with the number of layers.

[0036] Figure 4 This is a graph showing the iteration of Theta-ReduNet and ReduNet networks with the number of layers in an embodiment of the present invention.

[0037] Figure 5 This is a graph showing the KNN classification accuracy results of the network in this embodiment of the invention. Detailed Implementation

[0038] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0039] like Figure 2As shown, in one embodiment of the present invention, a normalized ReduNet white-box algorithm based on spherical coordinates includes the following steps:

[0040] S1. Obtain input data of sample size m in n-dimensional space, and transform the input data from Cartesian coordinate system to spherical coordinate system to obtain input data in spherical coordinate system;

[0041] S2. Set the radial coordinate to 1, define the input data in the spherical coordinate system on the unit sphere, and obtain the normalized data in the spherical coordinate system.

[0042] S3. Normalized data in spherical coordinates is used as the initial feature input to the ReduNet white-box network.

[0043] The expression for the input data in the spherical coordinate system in step S1 is:

[0044]

[0045] Where X represents the input data in spherical coordinates; x1 represents the 1D component of the input data in Cartesian coordinates; x2 represents the 2D component of the input data in Cartesian coordinates; x n-1 The input data consists of n-1 dimensional components in Cartesian coordinates; x n θ represents the n-dimensional components of the input data in Cartesian coordinates; θ1 represents the angular coordinates corresponding to the 1-dimensional components of the input data; θ n-2 θ represents the angular coordinates corresponding to the n-2 dimensional components of the input data. n-1 r represents the angular coordinates corresponding to the n-1 dimensional components of the input data; r is the radial coordinate.

[0046] In this embodiment, in most neural networks, all input samples are represented in Cartesian coordinates in n-dimensional space. Pre-activation from neurons is used to determine whether a neuron should be activated using the following activation function. In this process, the pre-activation is calculated using weights and bias parameters, represented in Cartesian coordinates. These Cartesian coordinates can be derived from spherical coordinates. Given a vector x represented in Cartesian coordinates in n-dimensional space, x can be defined in spherical coordinates θ = [r, θ], which consists of a radial coordinate r and n-1 angular coordinates θ = [θ1, θ2, ... θ]. n-1 Composed of [the following]. In this case, the k-th sample of x can be calculated from the formula:

[0047]

[0048] Generally, spherical coordinates are a three-dimensional version of polar coordinates. However, the spherical coordinates in the above equation represent all n-dimensional versions of polar coordinates, where n is greater than or equal to 2.

[0049] The expression for the normalized data in the spherical coordinate system in step S2 is:

[0050]

[0051] Where θ represents the normalized data in the spherical coordinate system; θ1 represents the angular coordinates corresponding to the 1-dimensional components of the input data; θ2 represents the angular coordinates corresponding to the 2-dimensional components of the input data; θ3 represents the angular coordinates corresponding to the 3-dimensional components of the input data; θ n-2 θ represents the angular coordinates corresponding to the n-2 dimensional components of the input data. n-1 x1 represents the angular coordinates corresponding to the (n-1)-dimensional components of the input data; x2 represents the 2-dimensional component of the input data in Cartesian coordinates; x3 represents the 3-dimensional component of the input data in Cartesian coordinates; x n-1 The input data consists of n-1 dimensional components in Cartesian coordinates.

[0052] In this embodiment, a popular assumption for high-dimensional classification problems is that each class has a relatively low-dimensional intrinsic structure. ReduNet aims to learn low-dimensional intrinsic representations of data by extracting features from samples of different classes belonging to different linear subspaces, while samples from the same class are highly correlated and belong to the same linear subspace. The objective function is constructed using a rate distortion function as a building block. Here, the application of sphere normalization in white-box networks is named Theta-ReduNet. From the spherical coordinate system, since each layer of the previous ReduNet network was always normalized by projecting onto a unit sphere, we define the radial coordinate r = 1 in the transformation between the spherical coordinate system and the Cartesian coordinate system of the n-dimensional data.

[0053] set up Let m be the input data matrix of sample size m, where each Let be the i-th n-dimensional sample, belonging to one of the k classes, and its corresponding spherical coordinate data matrix be... For a given precision ∈ > 0, Theta-ReduNet first transforms the input variables from a Cartesian coordinate system to a spherical coordinate system using the transformation between the spherical and Cartesian coordinate systems of the N-dimensional data. It then calculates the variance using the inverse formula of the formula for calculating the variance of the input data in layer normalization, and sets r = 1 to obtain the initial value of θ:

[0054]

[0055] In this embodiment, through the above data preprocessing, the initial values ​​of the input data for feature Z in the Cartesian coordinates of the first step of ReduNet are transformed into the initial values ​​of the input data for feature Z in spherical coordinates. By defining the radius r = 1, the data is always defined on the unit sphere, thus eliminating the need for the normalization operation after each feature iteration.

[0056] In step S3, the gradient of the ReduNet white-box network used to solve the maximum bitrate problem is:

[0057]

[0058]

[0059] Where R is the coding rate; The sign for the partial derivative is θ; θ represents the normalized data in spherical coordinates; θ 1 The normalized data is in the spherical coordinate system of the first sample; θ 2 The normalized data is in the spherical coordinate system of the second sample; θ m Z represents the normalized data of the m-th sample in spherical coordinates; m is the total number of samples; Z 1 Z represents the features of the first sample. 2 Features of the second sample; Z m G represents the features of the m-th sample; 1 G is the transpose of the derivative matrix of the first sample; 2 G is the transpose of the derivative matrix of the second sample; m G is the transpose of the derivative matrix of the m-th sample; T is the transpose; G i J is the transpose of the derivative matrix of the i-th sample; i Z is the derivative matrix for the i-th sample; i θ represents the features of the i-th sample; i θ1 represents the normalized data in the spherical coordinate system of the i-th sample; θ2 represents the angular coordinates corresponding to the 1-dimensional components of the input data; θ3 represents the angular coordinates corresponding to the 3-dimensional components of the input data; θ4 represents the normalized data in the spherical coordinate system of the i-th sample. p θ represents the angular coordinates corresponding to the p-dimensional components of the input data. n-1 The input data consists of n-1 dimensional components with angular coordinates; n is the spatial dimension; and i is the sample number.

[0060] In this embodiment, the transformation under the new spherical coordinate system is applied to the white-box network, and the data composition distribution and target iterative algorithm of the white-box network are re-inferred: after transforming the original Cartesian coordinate system to the spherical coordinate system, the features and iteration variables of the original network are changed from Z = [Z1, Z2, ... Z...]. m The result is θ = [θ1, θ2, ... θ].m Meanwhile, in white-box networks, the gradient objective for solving the maximum bit rate (MCR2) problem using gradient ascent is derived from... It became At this point, the formula can be obtained through verification using the chain rule of gradients:

[0061]

[0062] Furthermore, since the Z matrix is ​​composed of θ matrices, its partial derivatives with respect to θ matrices can be easily obtained, and the transpose of the derivative matrix J is denoted as the G matrix, thus we have the formula:

[0063]

[0064] At this point, the gradient objective for solving the maximum bit rate (MCR2) problem using gradient ascent in a white-box network becomes the following formula:

[0065]

[0066] The gradient ascent formula for the ReduNet white-box network in step S3 is as follows:

[0067]

[0068] Where, θ l+1 θ represents the angular coordinates of the (l+1)th layer of the network. l Let be the angular coordinates of the l-th layer network; η be the learning rate; and R be the encoding rate. is the sign of the partial derivative; θ represents the normalized data in spherical coordinates. For the case of the l-th layer network; Z 1 Z represents the features of the first sample. 2 Features of the second sample; Z m G represents the features of the m-th sample; 1 G is the transpose of the derivative matrix of the first sample; 2 G is the transpose of the derivative matrix of the second sample; m It is the transpose of the derivative matrix of the m-th sample.

[0069] In this embodiment, the feature iteration update step using gradient ascent in the white-box network becomes the following formula. Therefore, the network represents different data using only angles, and the network objective's iterative algorithm achieves the optimal solution by optimizing only the angles. The optimization formula for the entire network is:

[0070]

[0071] This is the known formula of the original ReduNet network. Therefore, the feature Z that originally needed to be updated and iterated is now simply the angle θ that needs to be updated, and the gradient ascent formula also becomes:

[0072]

[0073] As observed from the above formulas, the normalization method in spherical coordinates can eliminate the normalization to the unit sphere after each iteration. Furthermore, by using only angles to represent all representations in different ways, neural networks can reduce the complexity and speed of iteration.

[0074] Simulation test results:

[0075] In the simulation experiment, 7000 samples from the MNIST training set were used to train the original network and the angle-normalized network, while 3000 samples from the test set were used to evaluate the generalization error. All samples were preprocessed with dimensionality reduction n=87 and maximum normalization, with parameters chosen as η=0.5, λ=500, ∈=0.2, and δ=10⁻⁸. The experimental results are as follows. Figure 3 As shown.

[0076] like Figure 4 As shown in the simulation results, the loss curve of the modified network is displayed. It can be seen that in the final MCR2 objective, the loss increases with the number of layers and decreases with the number of layers. The objective achieves the maximized result and reaches the convergence effect of the experiment at around 800 layers, which is in line with the expected result.

[0077] Simulation results show a comparison of Theta-ReduNet and ReduNet networks with varying layer iterations. It can be seen that under the same parameter settings, Theta-ReduNet improves the model's numerical stability and convergence speed, achieving convergence more quickly. Table 1 illustrates a comparison of the image classification accuracy of Theta-ReduNet and ReduNet networks.

[0078] Table 1

[0079]

[0080] like Figure 5As shown, simulation results compare the classification accuracy of four different networks. The features extracted by Algorithm 1 can be used to determine which class a sample belongs to. This requires a suitable classifier, such as a Support Vector Machine, K-Nearest Neighbors, or Nearest Subspace. Here, the test accuracy of four networks is compared using the KNN classifier: the original ReduNet network, the AR-ReduNet network with bisection optimization, the Theta-ReduNet network (a spherical normalized network modified from the original ReduNet network), and the AR-ReduNet network (a spherical normalized network modified from the AR-ReduNet network). By comparing the classification accuracy evaluation metrics of the four sets of experiments, it can be observed that the original ReduNet network modified with spherical coordinates shows a significant improvement in accuracy compared to the original network. For a fixed distortion ∈ = 0.2, the Theta-ReduNet network outperforms the original network for all layers from L = 100 to 2000 and achieves an accuracy similar to AR-ReduNet. In addition, using spherical coordinate normalization on the more accurate AR-ReduNet network can further improve the performance. Experimental results reveal that spherical coordinate system normalization simplifies the original network, making it more suitable for the intrinsic structure of the data in some aspects, reducing certain noisy features of the data, and thus improving the final classification performance of the classifier.

[0081] Simulation Result Analysis: Analysis of the experimental results shows that using spherical coordinates to simplify the normalization of the original network improves classification accuracy. The possible reasons for this are as follows:

[0082] (1) Feature Extraction: Data distribution in spherical coordinates is easier to separate. Data that is non-linearly separable in Cartesian coordinates may exhibit linearly separable characteristics in spherical coordinates. This may better reflect the inherent structure of the data, making it easier for the model to learn the data patterns. Simultaneously, spherical coordinates may reduce redundant information in the data. In Cartesian coordinates, each component of the normalized vector Z is independent, but in high-dimensional space, there may be redundant information between these components. By converting to spherical coordinates, these points are represented in a more compact way because there are dependencies between the angular components in spherical coordinates. This representation reduces the redundant information that the model needs to learn, thereby improving classification accuracy. Spherical coordinate transformation can convert linearly inseparable features in the original data into linearly separable features. In some application scenarios, the distribution of the original data in Cartesian coordinates may be complex, making it difficult to classify effectively using a linear classifier. However, through spherical coordinate transformation, these complexly distributed data can be mapped to spherical coordinates, making the data distribution simpler and easier to classify.

[0083] (2) Data Distribution: A more uniform distribution of data in spherical coordinates may make certain classification algorithms more effective. Spherical coordinates may be better suited to the intrinsic structure of the data, making it easier to distinguish in a new feature space. In spherical coordinates, data may be more concentrated near the sphere, making it easier for classifiers to capture local features and improve classification accuracy. In some cases, the intrinsic structure of the data may be closer to spherical coordinates than Cartesian coordinates. For example, if data points are distributed on or near a unit sphere, then using spherical coordinates may be more natural to represent these data. In this case, spherical coordinate transformation may make it easier for the model to learn the intrinsic patterns of the data.

Claims

1. A normalized ReduNet white-box algorithm based on spherical coordinates, characterized in that, Includes the following steps: S1, Obtain n The sample size in 3D space is m The input data is obtained by converting the input data from the Cartesian coordinate system to the spherical coordinate system, resulting in input data in the spherical coordinate system; the input data is a grayscale image of a handwritten digit. S2. Set the radial coordinate to 1, and define the input data in the spherical coordinate system on a unit sphere to obtain the normalized data in the spherical coordinate system; the expression for the normalized data in the spherical coordinate system in step S2 is: in, Normalized data in spherical coordinates; The angular coordinates corresponding to the 1D components of the input data; The angular coordinates corresponding to the 2D components of the input data; The angular coordinates corresponding to the 3D components of the input data; For input data Angular coordinates corresponding to the dimensional components; For input data Angular coordinates corresponding to the dimensional components; The input data consists of one-dimensional components in Cartesian coordinates. The input data consists of two-dimensional components in Cartesian coordinates; The input data consists of 3D components in Cartesian coordinates; Input data in Cartesian coordinate system Dimensional components; S3. Normalized data in spherical coordinates is used as the initial feature input to the ReduNet white-box network.

2. The ReduNet white-box algorithm based on normalized spherical coordinates according to claim 1, characterized in that, The expression for the input data in the spherical coordinate system in step S1 is: in, Input data in spherical coordinate system; The input data consists of one-dimensional components in Cartesian coordinates. The input data consists of two-dimensional components in Cartesian coordinates; Input data in Cartesian coordinate system Dimensional components; Input data in Cartesian coordinate system n Dimensional components; The angular coordinates corresponding to the 1D components of the input data; For input data Angular coordinates corresponding to the dimensional components; For input data Angular coordinates corresponding to the dimensional components; For radial coordinates.

3. The ReduNet white-box algorithm based on normalized spherical coordinates according to claim 1, characterized in that, In step S3, the gradient of the ReduNet white-box network used to solve the maximum bitrate problem is: in, For coding rate; The sign for partial derivatives; Normalized data in spherical coordinates; This is the normalized data in the spherical coordinate system of the first sample; This is the normalized data in the spherical coordinate system of the second sample. For the first Normalized data in a spherical coordinate system for each sample; The total number of samples; The features of the first sample in Cartesian coordinates; Features of the second sample in Cartesian coordinates; For the first Features of a sample in Cartesian coordinates; This is the transpose of the derivative matrix of the first sample; This is the transpose of the derivative matrix of the second sample; For the first Transpose of the derivative matrix of each sample; For transpose; For the first Transpose of the derivative matrix of each sample; For the first The derivative matrix of each sample; For the first Features of a sample in Cartesian coordinates; For the first Normalized data in a spherical coordinate system for each sample; The angular coordinates corresponding to the 1D components of the input data; The angular coordinates corresponding to the 2D components of the input data; The angular coordinates corresponding to the 3D components of the input data; For input data Angular coordinates corresponding to the dimensional components; For input data Angular coordinates corresponding to the dimensional components; Spatial dimension; This is the sample number.

4. The ReduNet white-box algorithm based on normalized spherical coordinates according to claim 1, characterized in that, The gradient ascent formula for the ReduNet white-box network in step S3 is as follows: in, For the first Angular coordinates of the layer network; For the first Angular coordinates of the layer network; The learning rate; For coding rate; The sign for partial derivatives; Normalized data in spherical coordinates; In the first In the case of layered networks; The features of the first sample in Cartesian coordinates; Features of the second sample in Cartesian coordinates; For the first Features of a sample in Cartesian coordinates; This is the transpose of the derivative matrix of the first sample; This is the transpose of the derivative matrix of the second sample; For the first The transpose of the derivative matrix of each sample.

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