Multiview learning method, device and medium based on low rank constraint and hypergraph learning

Abstract

The application discloses a multi-view learning method and device based on low-rank constraint and hypergraph learning and a medium, and relates to the technical field of multi-view clustering.The method comprises the following steps: performing deep non-negative matrix decomposition on a data matrix of a multi-view to obtain a coefficient matrix; constructing a reconstruction error of the deep non-negative matrix decomposition through a norm; performing low-rank constraint on the coefficient matrix to obtain a low-rank constraint term; performing hypergraph structure learning on the coefficient matrix to obtain a hypergraph regularization term; establishing a target function of multi-view learning based on the reconstruction error, the low-rank constraint term and the hypergraph regularization term; and solving the target function to obtain a learning result of the multi-view. The low-rank constraint is used for the coefficient matrix, the sparsity of the coefficient matrix is enhanced, and redundant information is removed. The hypergraph structure is learned while the coefficient matrix is constructed, the multi-element relationship between data points is learned, the manifold structure information of a source space is retained, the expression of the coefficient matrix to deep information of data is enhanced, and the accuracy of a clustering result is improved.

Description

Technical Field

[0001] This invention relates to the field of multi-view clustering technology, and in particular to multi-view learning methods, devices and media based on low-rank constraints and hypergraph learning. Background Technology

[0002] In clustering, nonnegative matrix factorization (NMF) has become a popular method due to its simple structure and clear theory. NMF decomposes a high-dimensional data matrix X into a low-rank basis matrix W and a low-rank coefficient matrix H. The coefficient matrix H represents the relationships between data points in the original space X, and it has been proven to have excellent performance.

[0003] However, existing data is complex, often containing multi-level relationships and non-linear structures, making it difficult to fully extract information using only single-level non-negative matrix factorization. Therefore, to address this issue, deep non-negative matrix factorization (DNF) with a multi-level decomposition structure has been proposed. DNF is a multi-level matrix factorization method where the coefficient matrix H of the previous level is used in the decomposition of the next level, generating the basis matrix and coefficient matrix of the next level.

[0004] The widespread use of intelligent recognition and the prevalence of the internet have generated a large amount of high-dimensional, multi-source data. This data typically has multiple views, and different views often contain unique information. Due to the diversity of information between views, multi-view clustering methods that fully utilize this information are often superior to single-view clustering methods. To more effectively process multi-view data and improve clustering performance, numerous deep non-negative matrix factorization (DNF) methods based on multiple views have been proposed, including: adding a diversity regularization term to the deep matrix factorization to extract diversity information between views; constructing a consensus expression matrix to represent the consistency information between views to enhance clustering performance; constructing a redundant balancing term to balance the consistency and diversity information between views; adding weights to each layer of decomposition to reduce feature loss during decomposition; and using l 2,1 The Frobenius norm is replaced by a new norm to enhance clustering performance in noisy conditions; and partition alignment regularization terms are used to mine consistency information between views and fuse coefficient matrices from different views.

[0005] The multi-view depth matrix factorization methods described above have achieved good clustering results, but some problems still exist: First, current depth matrix factorization methods cannot guarantee the robustness of clustering under noisy conditions; second, most depth matrix factorization clustering methods ignore the information of local graphs or only consider the manifold structure of general graphs; finally, most depth matrix factorization methods retain a lot of redundant information when constructing the coefficient matrix. All of these problems affect the final clustering results. Summary of the Invention

[0006] This invention provides a multi-view learning method, device, and medium based on low-rank constraints and hypergraph learning to solve the problem that existing deep matrix factorization methods retain a lot of redundant information when constructing coefficient matrices, ignore local graph information, affect the expression of coefficient matrices, and ultimately reduce clustering results.

[0007] This invention is achieved through the following technical solution:

[0008] A first aspect of the present invention provides a multi-view learning method based on low-rank constraints and hypergraph learning, comprising:

[0009] Perform depthwise nonnegative matrix decomposition on the data matrix of the multi-view to obtain the coefficient matrix;

[0010] Constructing the reconstruction error of deep nonnegative matrix decomposition using norms;

[0011] Apply low-rank constraints to the coefficient matrix to obtain low-rank constraint terms;

[0012] Hypergraph structure learning is performed on the coefficient matrix to obtain the hypergraph regularization term;

[0013] A target function for multi-view learning is established based on the reconstruction error, low-rank constraint term, and hypergraph regularization term.

[0014] Solve the objective function to obtain the learning results of the multi-view.

[0015] The multi-view learning method of this invention, based on deep nonnegative matrix factorization, applies low-rank constraints and hypergraph structure learning to the coefficient matrix of the deep factorization. Low-rank constraints enhance the sparsity of the coefficient matrix and remove redundant information. Simultaneously with constructing the coefficient matrix, the hypergraph structure is learned, allowing for the study of multivariate relationships between data points while preserving the manifold structure information of the source space. This captures more local information between data points, thereby enhancing the coefficient matrix's representation of deep data information and improving the accuracy of clustering results.

[0016] Furthermore, the depthwise nonnegative matrix decomposition of the data matrix of the multiple views is expressed as:

[0017] X≈W1W2…W r …W R H R

[0018] in, For data matrix, Let m represent an m×n dimensional real matrix, where m represents the number of data features and n represents the number of samples. Let R represent the basis matrix of the r-th level decomposition, where r∈[0,R] and R is the total number of decomposition levels; Let represent the coefficient matrix of the R-th level decomposition, k << m.

[0019] Furthermore, the low-rank constraint term is expressed as:

[0020]

[0021] in, Let ||·|| represent the coefficient matrix of the R-th level decomposition of the v-th view. * V represents the nuclear norm number, and V is the total number of views.

[0022] Furthermore, the hypergraph regularization term is represented as:

[0023]

[0024] in, This represents the coefficient matrix of the R-th level decomposition of the v-th view. express The transpose of the matrix, Let represent the Laplacian matrix of the hypergraph of the v-th view, and Tr(·) represent the trace of the matrix.

[0025] Furthermore, the reflection is the arctangent norm, and the reconstruction error is expressed as:

[0026]

[0027] Among them, X v The data matrix representing the v-th view. This represents the basis matrix of the r-th level decomposition of the v-th view. Let ||·|| represent the coefficient matrix of the R-th level decomposition of the v-th view. arctan This represents the arctangent norm.

[0028] Furthermore, the optimization objective function is expressed as:

[0029]

[0030] st W r ≥0, H R ≥0, (r=1,2,…,R)

[0031] in, X represents the reconstruction error. v The data matrix representing the v-th view. This represents the basis matrix of the r-th level decomposition of the v-th view. Let ||·|| represent the coefficient matrix of the R-th level decomposition of the v-th view. arctan Represents the arctangent norm;

[0032] Denotes a low-rank constraint term, ||·|| * λ represents the nuclear norm, V is the total number of views, and λ1 is the parameter of the low-rank constraint term;

[0033] This represents the hypergraph regularization term. express The transpose of the matrix, Let λ2 represent the Laplacian matrix of the hypergraph of the v-th view, λ2 represent the parameters of the hypergraph regularization term, and Tr(·) represent the trace of the matrix.

[0034] st represents the constraint condition, r represents the number of decomposition levels, and R represents the total number of decomposition levels.

[0035] Furthermore, the optimization objective function is solved using an alternating iterative method, and the solution steps include:

[0036] S1, Fixed The value remains unchanged, update The update formula is:

[0037]

[0038] Where “.*” represents the Hadamard product of matrices, indicating the multiplication of corresponding elements of the matrices;

[0039] X v The data matrix representing the v-th view;

[0040] As an intermediate variable;

[0041]

[0042] for The transpose of the matrix, This represents the basis matrix of the r-th level decomposition of the v-th view. This represents the j-th column of the data matrix for the v-th view. express The j-th column of the matrix, for The transpose of the matrix, This represents the coefficient matrix of the R-th level decomposition of the v-th view;

[0043] S2, Fixed The value remains unchanged, update The update formula is:

[0044]

[0045] Where “.*” represents the Hadamard product of matrices, indicating the multiplication of corresponding elements of the matrices;

[0046] X v The data matrix representing the v-th view;

[0047] It is a diagonal matrix.

[0048] for The transpose of the matrix, This represents the basis matrix of the R-th level decomposition of the v-th view. This represents the j-th column of the data matrix for the v-th view. express The j-th column of the matrix, This represents the coefficient matrix of the R-th level decomposition of the v-th view;

[0049] M v express The left singular matrix of the singular value decomposition;

[0050] N v express The right singular matrix of singular value decomposition, (N) v ) T N represents v transpose;

[0051] express The similarity matrix of the hypergraph constraint;

[0052] express The degree matrix of the hypergraph constraint;

[0053] λ1 represents the parameter of the low-rank constraint term, and λ2 represents the parameter of the hypergraph regularization term;

[0054] S3: Substitute the updated parameters into the objective function for calculation, and determine whether the value of the objective function satisfies the convergence condition. If it does, stop the iteration; otherwise, repeat steps S1-S3.

[0055] A second aspect of the present invention provides a multi-view clustering method based on low-rank constraints and hypergraph learning, comprising:

[0056] The learning results of multiple views are obtained by using the multi-view learning method based on low-rank constraints and hypergraph learning as described in any one of the first aspects of the present invention.

[0057] Multi-view clustering is performed based on the learning results of the multi-views to obtain the clustering results.

[0058] A third aspect of the present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the multi-view learning method based on low-rank constraints and hypergraph learning as described in any of the first aspects of the present invention, and / or the multi-view clustering method based on low-rank constraints and hypergraph learning as described in the second aspect of the present invention.

[0059] A fourth aspect of the present invention provides a computer-readable storage medium having a computer program stored thereon, characterized in that, when the computer program is executed by a processor, it implements the multi-view learning method based on low-rank constraints and hypergraph learning as described in any one of the first aspects of the present invention, and / or the multi-view clustering method based on low-rank constraints and hypergraph learning as described in the second aspect of the present invention.

[0060] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0061] This invention uses low-rank constraints on the coefficient matrix to enhance its sparsity, remove redundant information, and solve the problem that most current deep matrix factorization methods retain a lot of redundant information when constructing the coefficient matrix.

[0062] This invention learns the hypergraph structure while constructing the coefficient matrix, learns the multivariate relationships between data points and retains the manifold structure information of the source space, solving the problem that most current deep matrix factorization clustering methods ignore the information of the local graph or only consider the manifold structure of the general graph.

[0063] This invention constructs the arctangent norm as the reconstruction error, which is more robust than the norm used in most current deep matrix factorization methods. 2，1 Norms and Frobenius norms improve clustering performance under various noise conditions. Attached Figure Description

[0064] To more clearly illustrate the technical solutions of the exemplary embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly described below. It should be understood that the following drawings only show some embodiments of the present invention and should not be considered as a limitation of the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort. In the drawings:

[0065] Figure 1 This is a flowchart of a multi-view learning method based on low-rank constraints and hypergraph learning according to an embodiment of the present invention;

[0066] Figure 2This is a flowchart illustrating a multi-view clustering method according to an embodiment of the present invention;

[0067] Figure 3 This is a schematic diagram of an electronic device structure according to an embodiment of the present invention. Detailed Implementation

[0068] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the embodiments and accompanying drawings. The illustrative embodiments and descriptions of the present invention are only used to explain the present invention and are not intended to limit the present invention.

[0069] It should be noted that the terms "comprising" and "having" and any variations thereof in the specification, claims, and accompanying drawings of this invention are intended to cover non-exclusive inclusion, for example, a process, method, system, product, or device that includes a series of steps or units is not necessarily limited to other steps or units inherent in the device.

[0070] The terminology used in the various embodiments of this application is for the purpose of describing particular embodiments only and is not intended to limit the various embodiments of this application. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Unless otherwise defined, all terms used herein (including technical and scientific terms) have the same meaning as commonly understood by one of ordinary skill in the art to which the various embodiments of this application pertain. The terms (such as those defined in a generally used dictionary) are to be interpreted as having the same meaning as in the context of the relevant technical field and are not to be interpreted as having an idealized or overly formal meaning, unless clearly defined in the various embodiments of this application.

[0071] This invention provides a multi-view learning method, device, and medium based on low-rank constraints and hypergraph learning, which is applicable to multi-view feature learning and clustering under various noise conditions. It is beneficial to enhance the coefficient matrix representation of deep non-negative matrix factorization and improve the accuracy of clustering results.

[0072] To help understand the technical content of this application, a brief introduction and analysis of the relevant prior art is provided below.

[0073] In the field of multi-view learning / clustering, most multi-view nonnegative matrix factorization methods use the classic Frobenius norm to measure reconstruction error. The classic nonnegative matrix factorization model is as follows:

[0074]

[0075] stW≥0,H≥0

[0076] in, For data matrix, Let m represent an m×n dimensional real matrix, where m represents the number of data features and n represents the number of samples. Represents the basis matrix after nonnegative matrix decomposition. Let represent the coefficient matrix after nonnegative matrix decomposition, where k << m. F The Frobenius norm is used to measure the reconstruction error, and E ij Let represent the element in the i-th row and i-th column of matrix E.

[0077] Nonnegative matrix factorization (NMF) uses a coefficient matrix H to represent the relationship between sample points and clusters. However, NMF only has one level of decomposition, making it difficult to uncover deeper information between data points. Therefore, deep nonnegative matrix factorization (DNF) was proposed. By decomposing the coefficient matrix into multiple layers, it learns complex and nonlinear relationships. The steps of deep NMF are as follows:

[0078] Level 1 decomposition: X≈W1H1;

[0079] Second-level decomposition: X≈W1W2H2;

[0080] ...

[0081] Decomposition at level R: X≈W1W2…W r …W R H R .

[0082] in, For data matrix, Let m represent an m×n dimensional real matrix, where m represents the number of data features and n represents the number of samples. Let R represent the basis matrix of the r-th level decomposition, where r∈[0,R] and R is the total number of decomposition levels; Let represent the coefficient matrix of the R-th level decomposition, k << m.

[0083] The objective function formula for deep nonnegative matrix factorization in the case of multi-view clustering is:

[0084]

[0085] in, Given a dataset containing V views, where mu represents the number of features in the v-th view, and n represents the number of samples, further... This represents the basis matrix of the first-level decomposition of the v-th view. This represents the basis matrix of the r-th level decomposition of the v-th view. Let l1 represent the coefficient matrix of the R-th level decomposition of the v-th view, and l1 represent the size of the 1-th level decomposition. r L represents the size of the r-th level decomposition. RIndicates the size of the last level of decomposition, ||·|| F This represents the Frobenius norm.

[0086] Deep nonnegative matrix factorization (DNF) addresses the challenge of fully uncovering multi-level relationships and nonlinear structures in complex data using only a single-layer DNF. However, most DNF clustering methods neglect information from local graphs or only consider the general graph manifold structure. Furthermore, they retain significant redundant information when constructing the coefficient matrix.

[0087] To address the aforementioned issues, this invention proposes a multi-view learning method based on low-rank constraints and hypergraph learning. Building upon deep nonnegative matrix factorization, the method performs low-rank constraints and learns the hypergraph structure of the coefficient matrix, thereby enhancing the representation of the coefficient matrix.

[0088] like Figure 1 The diagram shows a flowchart of a multi-view learning method based on low-rank constraints and hypergraph learning according to the present invention, which includes the following steps:

[0089] Step 1: Perform depth non-negative matrix decomposition on the data matrix of the multi-view to obtain the coefficient matrix.

[0090] Step 2: Construct the reconstruction error of deep nonnegative matrix factorization through norm; apply low-rank constraints to the coefficient matrix to obtain low-rank constraint terms; and perform hypergraph structure learning on the coefficient matrix to obtain hypergraph regularization terms.

[0091] Step 3: Based on the reconstruction error, low-rank constraint term, and hypergraph regularization term obtained in Step 2, establish the objective function for multi-view learning.

[0092] Step 4: Solve for the objective function to obtain the learning results of the multi-view.

[0093] The learning results of the multi-view system include the basis matrix and the solution values ​​corresponding to the coefficient matrix obtained by deep nonnegative matrix decomposition in step 1.

[0094] In step 1, the deep nonnegative matrix factorization method is used to perform deep decomposition on the data matrix of the multi-view. The basis matrix and coefficient matrix obtained after R-level decomposition can be expressed as:

[0095] X≈W1W2…W r …W R H R

[0096] in, A data matrix with multiple views. Let m represent an m×n dimensional real matrix, where m represents the number of data features and n represents the number of samples. Let R represent the basis matrix of the r-th level decomposition, where r∈[0,R] and R is the total number of decomposition levels; Let represent the coefficient matrix of the R-th level decomposition, k << m.

[0097] During the learning process, the aforementioned depthwise nonnegative matrix decomposition is performed on each view v in the multi-view data to obtain the basis matrix W of each layer. r and the final coefficient matrix H R .

[0098] In step 2, the coefficient matrix H R Establishing low-rank constraints can enhance the sparsity of sparse matrices and remove redundant information. The low-rank constraint term can be expressed as:

[0099]

[0100] in, Let ||·|| represent the coefficient matrix of the R-th level decomposition of the v-th view, where V represents the total number of views. * Denotes the nuclear norm, which is The sum of all singular values ​​obtained from singular value decomposition.

[0101] The formula for singular value decomposition is as follows:

[0102]

[0103] Among them, M v express The left singular matrix of singular value decomposition, ∑ v Let N represent a diagonal matrix composed of singular values. v express The right singular matrix of singular value decomposition, (N) v ) T N represents v The transpose of .

[0104] After incorporating low-rank constraints, the formula for multi-view depth non-negative matrix decomposition becomes:

[0105]

[0106] st W r ≥0, H R ≥0, (r=1,2,…,R)

[0107] in, Represents the reconstruction error, ||·|| F It is the Frobenius norm. This represents the basis matrix of the r-th level decomposition of the v-th view; Denotes a low-rank constraint term, ||·|| *Let λ represent the nuclear norm, λ1 represent the parameter of the low-rank constraint term, and st represent the constraint condition.

[0108] Most deep matrix factorization clustering methods neglect information from local graphs or only consider the manifold structure of general graphs. When mining structural information between general graphs, the formula is expressed as:

[0109]

[0110] Among them, L v This represents the Laplace matrix of the graph for the v-th view. Let be the similarity matrix, and let represent the coefficient matrix. The similarity relationship between paired points. Let be a degree matrix, where i represents the i-th row of the matrix, and j represents the j-th column of the matrix. This graph learning method connects only two points per edge, therefore it can only mine binary relationships between data points.

[0111] This invention learns the hypergraph structure while constructing the coefficient matrix, learns the multivariate relationships between data points, and preserves the manifold structure information of the source space. Considering the hypergraph structure in all views, the hypergraph regularization term is expressed as:

[0112]

[0113] in, This represents the coefficient matrix of the R-th level decomposition of the v-th view. express The transpose of the matrix, Let represent the Laplacian matrix of the hypergraph of the v-th view, and Tr(·) denote the transpose of the matrix.

[0114] The Laplacian matrix of the hypergraph of the v-th view:

[0115]

[0116] Among them, H v W represents the matrix W that represents the relationships between vertices and hyperedges in the hypergraph of the v-th view. v It is a weight matrix. Let be the degree matrix of the vertices. Let be the degree matrix of the hyperedge. for The inverse matrix, (H v ) T For H v The transpose of .

[0117] The reconstruction error of deep nonnegative matrix factorization (DNF) is constructed using norms. Standard DNF uses the Frobenius norm to represent the reconstruction error; however, the Frobenius norm is sensitive to noise. To improve robustness, some DNF methods employ l... 2，1 The norm is used to represent the reconstruction error, and the formula is as follows:

[0118]

[0119] Among them, ||·|| 2，1 Indicate l 2，1 Norm, ||E|| 2，1 =∑ j ||E(:,j)||2, where E(:,j) is the j-th column of matrix E, and st represents the constraint condition. 2，1 Norms can improve the robustness of clustering.

[0120] Then based on l 2，1 The reconstruction error of the norm-based deep nonnegative matrix factorization, combined with low-rank constraints and hypergraph structure learning, yields the following formula for multi-view deep nonnegative matrix factorization:

[0121]

[0122] l 2，1 Norms improve the robustness of clustering, but still have two drawbacks: First, l 2，1 The norm value remains relatively large even under high noise conditions; secondly, l 2，1 The loss function of the norm has no upper bound, which means that larger outliers will have a greater impact on clustering.

[0123] Therefore, the present invention preferably uses the arctangent norm ||·|| arctan This is used to represent the reconstruction error.

[0124] In high-noise conditions, the arctangent norm value is less than l. 2，1 The arctangent norm; secondly, the arctangent norm has an upper bound of π / 2. When using the arctangent function to represent the reconstruction error of depth matrix decomposition, the formula can be expressed as:

[0125]

[0126] Among them, the arctangent norm in Let E(j) be the reconstruction error matrix of the v-th view, and let E(j) represent the reconstruction error vector corresponding to the j-th sample.

[0127] The objective function for heat generation multi-view learning, based on reconstruction error, low-rank constraint term, and hypergraph regularization term, can be expressed as:

[0128]

[0129] st W r ≥0, H R ≥0, (r=1,2,…,R)

[0130] in, X represents the reconstruction error. v The data matrix representing the v-th view. This represents the basis matrix of the r-th level decomposition of the v-th view. Let ||·|| represent the coefficient matrix of the R-th level decomposition of the v-th view. arctan Represents the arctangent norm;

[0131] Denotes a low-rank constraint term, ||·|| * λ represents the nuclear norm, V is the total number of views, and λ1 is the parameter of the low-rank constraint term;

[0132] This represents the hypergraph regularization term. express The transpose of the matrix, Let λ2 represent the Laplacian matrix of the hypergraph of the v-th view, λ2 represent the parameters of the hypergraph regularization term, and Tr(·) represent the trace of the matrix.

[0133] st represents the constraint condition, r represents the number of decomposition levels, and R represents the total number of decomposition levels.

[0134] In step 4, an alternating iterative method is used, meaning that when updating one variable, all other variables are kept fixed. Through multiple rounds of iteration, the algorithm converges, yielding the final values ​​of each variable. The solution steps include:

[0135] S1, Fixed The value remains unchanged, update

[0136] The basis matrix of the r-th level decomposition for the v-th view The update formula is expressed as:

[0137]

[0138] st W r ≥0

[0139] in,

[0140] When using e j When representing the reconstruction error of the j-th sample, the above... The update formula can be written in general form:

[0141]

[0142] When there are c variables [g1, g2, ..., gc], taking the partial derivative of the above equation and setting it to 0, the partial derivative formula is expressed as:

[0143]

[0144] set up The above partial derivative formula can be transformed into:

[0145]

[0146] In the formula, f(e j =arctan(||e j ||2) and

[0147] Therefore, the solution can be found The problem can be rephrased as:

[0148]

[0149] st W r ≥0

[0150] in, This represents the j-th column of the v-th view data matrix. This represents the basis matrix of the r-th level decomposition of the v-th view. express The j-th column of the matrix, ||·||2 represents the L2 norm,

[0151] Furthermore, solve The question can be written as:

[0152]

[0153] st W r ≥0

[0154] Here,

[0155] Rewritten in augmented Lagrange form, we get:

[0156]

[0157] Where Φ is a Lagrange multiplier.

[0158] Considering the complementary relaxation in the Karush-Kuhn-Tucker (KKT) conditions, we can obtain:

[0159]

[0160] right Taking the partial derivative, we get:

[0161]

[0162] set up Taking complementary relaxation into account, we can obtain The final update formula is:

[0163]

[0164] Where “.*” represents the Hadamard product of matrices, indicating the multiplication of corresponding elements of the matrices;

[0165] X v The data matrix representing the v-th view;

[0166]

[0167] for The transpose of the matrix, This represents the basis matrix of the r-th level decomposition of the v-th view. This represents the j-th column of the data matrix for the v-th view. express The j-th column of the matrix, for The transpose of the matrix, This represents the coefficient matrix of the R-th level decomposition of the v-th view.

[0168] (2) Fixed The value remains unchanged, update

[0169] When fixed At that time, for the coefficient matrix of the R-th level decomposition of the v-th view The update formula can be expressed as:

[0170]

[0171] st H R ≥0

[0172] In the formula, With Update The formula is similar. The update formula can be expressed as:

[0173]

[0174] st H R ≥0

[0175] In the formula,

[0176]

[0177] in, for The transpose of the matrix, This represents the basis matrix of the R-th level decomposition of the v-th view. This represents the j-th column of the data matrix for the v-th view. express The j-th column of the matrix, This represents the coefficient matrix of the R-th level decomposition of the v-th view.

[0178] Rewritten in augmented Lagrange form, we get:

[0179]

[0180] Where Λ represents the Lagrange multiplier, yes The kernel function is equal to Singular Value Decomposition The sum of all singular values ​​obtained. Although the kernel function It is not differentiable, but it still has a subgradient M. v (N v ) T .

[0181] right Taking the partial derivative, we can obtain

[0182]

[0183] Here [M v (N v ) T ] + and [M] v (N v ) T ] - M respectively v (N v ) T The positive and negative parts of the value.

[0184] and solving The derivation is similar. The final update formula is:

[0185]

[0186] Where “.*” represents the Hadamard product of matrices, indicating the multiplication of corresponding elements of the matrices;

[0187] X v The data matrix representing the v-th view;

[0188] It is a diagonal matrix.

[0189] for The transpose of the matrix, This represents the basis matrix of the R-th level decomposition of the v-th view. This represents the j-th column of the data matrix for the v-th view. express The j-th column of the matrix, This represents the coefficient matrix of the R-th level decomposition of the v-th view;

[0190] M v express The left singular matrix of the singular value decomposition;

[0191] N v express The right singular matrix of singular value decomposition, (N) v ) T N represents v transpose;

[0192] express The similarity matrix of the hypergraph constraint;

[0193] express The degree matrix of the hypergraph constraint;

[0194] λ1 represents the parameter of the low-rank constraint term, and λ2 represents the parameter of the hypergraph regularization term.

[0195] Finally, the convergence criteria are set as follows:

[0196]

[0197] Among them, obj (k) is the value of the objective function in the k-th iteration, ∈ is a small positive parameter. After each iteration, it is checked whether the convergence condition has been met. If the convergence condition is met, the iteration stops. If not, the next step (1)-(2) is executed.

[0198] By solving the objective function, the basis matrix of the final depth nonnegative matrix decomposition is obtained. sum coefficient matrix The clustering features of multiple views are learned. Further clustering of multiple views can be performed based on the learned results to obtain the clustering outcome.

[0199] A second aspect of the present invention provides a multi-view clustering method based on low-rank constraints and hypergraph learning, such as... Figure 2 The diagram shown is a flowchart of the multi-view clustering method, including:

[0200] S1, the multi-view learning result is obtained by using the multi-view learning method based on low-rank constraints and hypergraph learning as described in any of the above embodiments of the present invention;

[0201] S2, perform multi-view clustering based on the learning results of the multi-view, and obtain the clustering results.

[0202] Specifically, step S1 combines Figure 2 As shown, and referring to steps 1 to 4 of the above embodiments of the present invention, the multi-view learning result includes the basis matrix of the deep nonnegative matrix factorization. sum coefficient matrix pass Obtain the fusion coefficient matrix And on Use k-means to obtain the clustering results.

[0203] The robustness of the proposed method in clustering was verified using the COIL20 dataset. Poisson noise, multiplicative noise, Gaussian noise, salt-and-pepper noise, and random occlusion noise were added to the COIL20 dataset to generate corresponding noisy datasets. The Gaussian noise used had a normalization deviation of 0.2, the salt-and-pepper noise intensity was defined as pixels with 20% contamination, and the random occlusion size was defined as 48×48 pixels. The proposed method was compared with other methods on various noisy datasets to compare the clustering performance of different methods under noisy conditions. Experiments were performed 10 times on each dataset, and the mean and standard deviation of each metric were recorded. Six common clustering performance metrics were used: Accuracy (ACC), Normalized Mutual Information (NMI), Purity, F-score, Precision, and Adjusted Land Coefficient (AR). For these six metrics, higher values ​​indicate better clustering results. The results are shown in Table 1, with bolded values ​​indicating the optimal results. Table 1 shows that under different noise conditions, the proposed method generally outperforms the comparative methods in all six metrics. Therefore, the proposed method has stronger robustness.

[0204] Table 1. ACC values ​​for various methods under different noise intensities.

[0205]

[0206]

[0207] Furthermore, the proposed method was implemented on six multi-view datasets: BBCsport, BBC4view, NGs, COIL20, 3Sources, and Citeseer. Brief information about these datasets is provided in Table 2. The experimental results of this invention utilize six common clustering performance metrics: Accuracy (ACC), Normalized Mutual Information (NMI), Purity, F-score, Precision, and Adjusted Land Coefficient (AR). For these six metrics, higher values ​​indicate better clustering results. To demonstrate the overall performance of the proposed method, it was compared with a single-view depth matrix factorization method (DSNMF), a multi-view nonnegative matrix factorization method (RMNMFGD), and two multi-view depth matrix factorization methods (AMDMD and MVDMFPA). The results are shown in Table 3.

[0208] Table 2 Multi-view Dataset Information

[0209]

[0210]

[0211] Table 3 compares the clustering performance of our method with several clustering methods based on nonnegative matrix factorization.

[0212]

[0213]

[0214] Table 3 shows that, across most datasets and most metrics, this method outperforms other methods in clustering. DSNMF is a single-view depth matrix factorization method, and its clustering performance is lower than that of multi-view methods. Multi-view methods can fully utilize information from each view, giving them an advantage in information diversity compared to single-view methods. RMNMFGD is l 2,1 Norm-constrained nonnegative matrix factorization (NMFGD), being a single-layer decomposition method, cannot fully extract deep information between data points. Therefore, RMNMFGD performs worse in clustering than the method proposed in this invention, which uses multi-layer decomposition. AMDMD and MVDMFPA are both multi-view deep matrix factorization methods. Compared to the method in this invention, these two methods ignore the multivariate manifold relationships between data points and fail to remove redundant information. Therefore, the method in this invention outperforms these two methods in clustering performance.

[0215] The multi-view learning method based on low-rank constraints and hypergraph learning proposed in this invention has the following advantages for multi-view clustering:

[0216] 1. This invention uses the arctangent norm instead of the Frobenius norm or l 2,1 The norm is used to measure the reconstruction error, which enhances the robustness performance under noisy conditions.

[0217] 2. This invention uses a matrix kernel function to apply a low-rank constraint to the coefficient matrix, thereby enhancing its sparsity and removing redundant information.

[0218] 3. This invention introduces a hypergraph structure to represent the manifold structure of the local graph, learns the multivariate relationships between data points and preserves the manifold structure information of the source space.

[0219] The key points mentioned above can be applied to similar products or other products. Here, "similar products" refers to methods related to multi-view clustering, and "other products" refers to methods in the field of machine learning other than clustering.

[0220] A third aspect of the present invention provides an electronic device, such as... Figure 3 As shown, Figure 3 This is a schematic diagram of the structure of an electronic device according to the present invention. The electronic device includes a processor 30, a memory 31, an input device 32, an output device 33, and a communication device 34; the number of processors 30 in the computer device can be one or more, to... Figure 3 Taking a processor 30 as an example; the processor 30, memory 31, input device 32, and output device 33 in an electronic device can be connected via a bus or other means. Figure 3 Taking the example of a connection between China and Israel via a bus.

[0221] The memory 31, as a computer-readable storage medium, can be used to store software programs, computer-executable programs, and modules. The processor 30 executes various functional applications and data processing of the electronic device by running the software programs, instructions, and modules stored in the memory 31, thereby implementing the multi-view learning method based on low-rank constraints and hypergraph learning, and / or the multi-view clustering method based on low-rank constraints and hypergraph learning, as described in any of the above embodiments of the present invention.

[0222] The memory 31 may primarily include a program storage area and a data storage area. The program storage area may store the operating system and at least one application program required for a given function; the data storage area may store data created based on terminal usage. Furthermore, the memory 31 may include high-speed random access memory and non-volatile memory, such as at least one disk storage device, flash memory, or other non-volatile solid-state storage device. In some instances, the memory 31 may further include memory remotely located relative to the processor 30, which can be connected to the electronic device via a network. Examples of such networks include, but are not limited to, the Internet, intranets, local area networks, mobile communication networks, and combinations thereof.

[0223] The input device 32 can be used to receive external data such as user ID, instructions, and multi-view data, while the output device 33 is used to output intermediate data, solution results, clustering results, and distribution network pages.

[0224] A fourth aspect of the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements a multi-view learning method based on low-rank constraints and hypergraph learning, and / or a multi-view clustering method based on low-rank constraints and hypergraph learning, according to any embodiment of the present invention. The storage medium may be ROM / RAM, magnetic disk, optical disk, etc.

[0225] In a fifth aspect, the present invention provides a computer program product that, when run on a computer, causes the computer to execute the multi-view learning method based on low-rank constraints and hypergraph learning, and / or the multi-view clustering method based on low-rank constraints and hypergraph learning, according to any of the above embodiments of the present invention.

[0226] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A multi-view learning method based on low-rank constraints and hypergraph learning, characterized in that, include: Perform depthwise nonnegative matrix decomposition on the data matrix of the multi-view to obtain the coefficient matrix; Constructing the reconstruction error of deep nonnegative matrix decomposition using norms; Apply low-rank constraints to the coefficient matrix to obtain low-rank constraint terms; Hypergraph structure learning is performed on the coefficient matrix to obtain the hypergraph regularization term; A target function for multi-view learning is established based on the reconstruction error, low-rank constraint term, and hypergraph regularization term. Solve the objective function to obtain the learning results of the multi-view; The depthwise nonnegative matrix decomposition of the data matrix of the multi-view is expressed as: ； in, For data matrix, express m × n A 3D real matrix m Represents the number of data features. n Indicates the number of samples; Indicates the first r The basis matrix of the layer decomposition, r The number of decomposition layers, r =1,2,⋯, R , R To decompose the total number of levels; Indicates the first R The coefficient matrix of the layer decomposition, ; The objective function is expressed as: ； ； in, Indicates reconstruction error, Indicates the first v Data matrix of views, Indicates the first v The first view r The basis matrix of the layer decomposition, Indicates the first v The first view R The coefficient matrix of the layer decomposition, Represents the arctangent norm; This represents a low-rank constraint term. Represents the nuclear norm. V Total number of views These are the parameters for the low-rank constraint terms; This represents the hypergraph regularization term. express The transpose of the matrix, Indicates the first v The Laplacian matrix of a hypergraph for each view. The parameter represents the hypergraph regularization term. Tr ( · ) represents the trace of a matrix; st These are constraints.

2. The multi-view learning method based on low-rank constraints and hypergraph learning according to claim 1, characterized in that, The objective function is solved using an alternating iterative method, and the solution steps include: S1, Fixed The value remains unchanged, update , The update formula is: ； Where ".*" represents the Hadamard product of matrices, indicating the multiplication of corresponding elements of the matrices; Indicates the first v Data matrix of views; , which is an intermediate variable; ； , for The transpose of the matrix, Indicates the first v The first view r The basis matrix of the layer decomposition, Indicates the first v The j-th column of the data matrix of each view express The first of the matrix j List, , for The transpose of the matrix, Indicates the first v The first view R The coefficient matrix of the layer decomposition; S2, Fixed The value remains unchanged, update , The update formula is: ； in, It is a diagonal matrix. ; ; , for The transpose of the matrix, Indicates the first v The first view R The basis matrix of the layer decomposition, Indicates the first v The j-th column of the data matrix of each view express The first of the matrix j List, Indicates the first v The first view R The coefficient matrix of the layer decomposition; express The left singular matrix of the singular value decomposition; express The right singular matrix of singular value decomposition, express transpose; express The similarity matrix of the hypergraph constraint; express The degree matrix of the hypergraph constraint; The parameter representing the low-rank constraint term. Parameters representing the hypergraph regularization term; S3: Substitute the updated parameters into the objective function for calculation, and determine whether the value of the objective function satisfies the convergence condition. If it does, stop the iteration; otherwise, repeat steps S1-S3.

3. A multi-view clustering method based on low-rank constraints and hypergraph learning, characterized in that, include: The learning results of multiple views are obtained by using the multi-view learning method based on low-rank constraints and hypergraph learning as described in any one of claims 1-2. Multi-view clustering is performed based on the learning results of the multi-views to obtain the clustering results.

4. An electronic device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the multi-view learning method based on low-rank constraints and hypergraph learning as described in any one of claims 1-2, and / or the multi-view clustering method based on low-rank constraints and hypergraph learning as described in claim 3.

5. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the multi-view learning method based on low-rank constraints and hypergraph learning as described in any one of claims 1-2, and / or the multi-view clustering method based on low-rank constraints and hypergraph learning as described in claim 3.

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