CN114254703A - Robust local and global regularization non-negative matrix factorization clustering method - Google Patents

Robust local and global regularization non-negative matrix factorization clustering method Download PDF

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CN114254703A
CN114254703A CN202111563605.8A CN202111563605A CN114254703A CN 114254703 A CN114254703 A CN 114254703A CN 202111563605 A CN202111563605 A CN 202111563605A CN 114254703 A CN114254703 A CN 114254703A
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张�杰
左芙蓉
张煜凡
向鹏宇
高伟
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Abstract

The invention relates to the technical field of data processing, in particular to a robust local and global regularized nonnegative matrix factorization clustering method, which comprises the following steps: acquiring an image clustering sample; constructing a nearest neighbor map on local scattering of a sample and introducing smooth regularization; representing the global geometry of the space by using transformation, and taking the global geometry as an additional main component graph regularization item to be incorporated into an NMF algorithm; applying graph regularization term constraints to an original NMF model by joint modeling and utilizing LPThe smoothness constraint restrains the base matrix; using correlation entropy to replace Euclidean norm in the error measurement, thereby obtaining a robust target function of local and global regularization nonnegative matrix decomposition; iterating preset times by using an iterative weighting method according to the target function, updating the variable U, V, and completing robust local and global regularized nonnegative matrix decomposition; using K-means clustering algorithm to pair coefficient momentsThe arrays are clustered.

Description

Robust local and global regularization non-negative matrix factorization clustering method
Technical Field
The invention relates to the technical field of data processing, in particular to a robust local and global regularized nonnegative matrix factorization clustering method.
Background
With the development of computer technology, high-dimensional data has been applied to different fields, and people pay more and more attention to data dimension reduction. The dimensionality reduction has wide application, and the single image is converted into a data set in a high-dimensional space through high-dimensionality of the single image data, so that the internal structure and information of multivariate data can be revealed and the multivariate data can be used for subsequent tasks such as visualization, classification and clustering.
non-Negative Matrix Factorization (NMF) has been frequently applied in the fields of pattern recognition, computer vision, and information inspection, as an effective dimension reduction method. The basic idea of non-negative matrix factorization is to find two low-dimensional non-negative matrices to approximate the original high-dimensional matrix, i.e. the original data matrix is reconstructed only by addition operation, which also makes the non-negative matrix factorization have the advantage of being wholly based on partial representation, and the non-negative matrix factorization has become one of the most powerful methods for clustering and feature selection. In order to improve the performance of the original NMF, researchers have developed various NMF extension methods from different aspects, for example, the multiplier alternating direction method (ADMM) is used to optimize the NMF method; using graph regularization nmf (gnmf), the intrinsic geometry of the data space is preserved by constructing a simple graph to account for the pairwise geometric relationships between samples; manifold regularization discrimination NMF (MD-NMF) considers the geometry of the data and the discrimination information … … for the different classes
The above methods all use euclidean norms to minimize the distance between the original data matrix and the reconstructed matrix. However, many data in the real world include gaussian noise, non-gaussian noise (for example, in the process of measuring and collecting gene expression data), or abnormal values, and in practical applications, it is a troublesome problem to effectively process the noise or abnormal values. In this case, the performance of the euclidean norm based non-negative matrix factorization method will be greatly reduced. The correlation entropy is proposed for robust analysis in information theory learning, has been proven to be effective in processing noise and outliers, and is widely applied in the fields of signal processing, biological information, face recognition, and the like. The correlation entropy is a measure of non-linearity and local similarity, the variation of which is related to the probability of similarity of two random variables. Instead of considering only the euclidean norm of the second moment, the correlation entropy may consider a higher order matrix.
Disclosure of Invention
Aiming at the problems, the invention discloses a robust local and global regularized non-negative matrix factorization clustering method, which adds related entropy into an objective function to effectively reduce the influence of noise and abnormal values, and improves the robustness of the robust local and global regularized non-negative matrix factorization clustering method on the noise and the abnormal values; in addition, the method also considers the geometric information of the data by combining a graph regularization term and utilizes LPSmoothness constraints constrain the base matrix to obtain a smooth and more accurate solution.
The invention adopts the following specific technical scheme:
a robust local and global regularized non-negative matrix factorization clustering method comprises the following steps:
s1, acquiring image clustering samples;
s20, constructing a nearest neighbor map on local scattering of the sample and introducing smooth regularization;
s30, representing the global geometry of the space by using transformation, and taking the global geometry as an additional main component graph regularization item to be included in an NMF algorithm;
s40 applying graph regularization term constraints to the original NMF model through joint modeling and utilizing LPThe smoothness constraint restrains the base matrix;
s50, using the correlation entropy to replace the Euclidean norm in the error measurement, thereby obtaining the robust target function of the local and global regularization nonnegative matrix decomposition;
s60, according to the objective function, iterating preset times by using an iterative weighting method, updating a variable U, V, and completing robust local and global regularization nonnegative matrix decomposition;
s70, performing clustering analysis on the coefficient matrix by adopting a K-means clustering algorithm. Compared with the traditional clustering method, the clustering method can more effectively reveal the internal geometric structure and the identification structure of the data, and improves the clustering performance.
Further preferably, in step S20, a nearest neighbor map is constructed on the local scattering of the sample, and a smooth regularization is introduced, where the smooth regularization specifically is:
Figure BDA0003421528340000021
where Tr (.) represents the trace of the matrix. L is the laplace matrix of the graph, L-D-W, D denotes a diagonal matrix, and each entry in D is WijIs the sum of the rows (or columns, since W is symmetrical), i.e. Dii＝∑jWij. W is a weight matrix, the element W in the matrixijThe definition is as follows:
Figure BDA0003421528340000031
wherein
Figure BDA0003421528340000032
Representing a set of K adjacent data points, X ═ X1,x2,...,xnIs a set of vertices.
Further preferably, the global geometry of the space is represented by a transformation at step S30 and is included as an additional principal component graph regularization term in the NMF algorithm, specifically, the global scattering on the coding matrix is maximized and defined as:
Figure BDA0003421528340000033
wherein,
Figure BDA0003421528340000034
representing the average of the encoded vectors.
The equation is further simplified and yields:
Figure BDA0003421528340000035
where M ═ I-E is called principal component diagram, E ═ 1/n) eeTI is an n × n identity matrix and e is an n-dimensional column vector with elements equal to 1.
Further preferably, the graph regularization term constraints are applied to the original NMF model by joint modeling and utilizing L at step S40PThe smoothness constraint is used for constraining the base matrix, and the obtained objective function of the non-negative matrix decomposition is specifically as follows:
Figure BDA0003421528340000036
where alpha and beta are two trade-off parameters.
By means of LPThe smoothness constraint constrains the base matrix to obtain a smooth and more accurate solution, specifically:
Figure BDA0003421528340000037
where λ is a non-negative parameter.
Further preferably, in step S50, the correlated entropy is used in place of the euclidean norm in the error metric, so as to obtain the objective function of the robust local and global regularized non-negative matrix factorization, which is specifically:
Figure BDA0003421528340000041
the first term of the equation is based on entropy reconstruction error, the second term is a local smoothness graph regular term, the third term represents a global geometry structure graph regular term, and the fourth term utilizes LPSmoothness constraints constrain the base matrix.
Further preferably, in step S60, iterating for a preset number of times by using an iterative weighting method according to the objective function, updating the variable U, V, and completing the non-negative matrix component of robust local and global regularization, the method includes:
establishing a Lagrangian function L according to the objective function of the non-negative matrix factorization:
L＝Tr(XXT)-2Tr(XVUT)+Tr(UVTVUT)
+αTr(VLVT)-βTr(VMVT)+2λ||U||P
+Tr(ΨUT)+Tr(ΦVT)
wherein Ψ ═ ψik]，Φ＝[φjk]；
Respectively solving the partial derivatives of the basic matrix U and the coefficient matrix V by using the Karush-Kuhn-Tucker condition
Figure BDA0003421528340000042
φjkνjk0 yields the iterative equations of the basis matrix U and the coefficient matrix V, respectively.
The update rule of the variable U is as follows:
Figure BDA0003421528340000043
the update rule of the variable V is as follows:
Figure BDA0003421528340000044
further preferably, the matrix U and the matrix V are separately subjected to partial derivation in step (step) using the Karush-Kuhn-Tucker condition
Figure BDA0003421528340000045
φjkνjkObtaining an iterative equation of the basis matrix U and the coefficient matrix V as 0, further including:
performing loop iteration according to the initialization weight, the basic matrix U and the iterative expression of the coefficient matrix V;
and after the cycle reaches the preset iteration times t, outputting a basic matrix U and a coefficient matrix V to finish the robust local and global regularized nonnegative matrix decomposition.
The invention has the beneficial effects that: compared with the traditional image discrimination method, the method adds the related entropy into the target function, is beneficial to capturing the high-order moment of the data, effectively reduces the influence of noise and abnormal values, and enhances the robustness of nonnegative matrix decomposition; the invention considers the geometric information of the data by combining the graph regularization term and utilizes LPThe smoothness constraint restrains the base matrix, so that a smooth and more accurate solution is obtained, and better identification precision is achieved; in addition, the present invention requires little computation time in terms of computational complexity to achieve the highest recognition accuracy on the database.
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FIG. 1 is a schematic flow chart of the present invention.
Detailed Description
For the purpose of enhancing the understanding of the present invention, the present invention will be described in further detail with reference to the accompanying drawings and examples, which are provided for the purpose of illustration only and are not intended to limit the scope of the present invention.
As shown in FIG. 1, a robust local and global regularized non-negative matrix factorization clustering method, which is based on graph regularized non-negative matrix factorization, includes the following steps:
step S1: acquiring an image clustering sample;
step S20: constructing a nearest neighbor map on the local scatter of the sample obtained in the step S1 and introducing smooth regularization;
step S30: representing the global geometry of the space by using transformation, and taking the global geometry as an additional main component graph regularization item to be incorporated into an NMF algorithm;
step S40: applying graph regularization term constraints to an original NMF model by joint modeling and utilizing LPThe smoothness constraint restrains the base matrix;
step S50: using correlation entropy to replace Euclidean norm in the error measurement, thereby obtaining a robust target function of local and global regularization nonnegative matrix decomposition;
step S60: iterating preset times by using an iterative weighting method according to the target function, updating the variable U, V, and completing robust local and global regularized nonnegative matrix decomposition;
step S70: and (5) carrying out clustering analysis on the coefficient matrix by adopting a K-means clustering algorithm.
The image data can be regarded as given an m × n matrix X ═ X1,x2,…,xn]∈Rm×nWherein x isi(i e {1,2, …, n }) represents a sample of the data, and the NMF method aims to decompose the data matrix X into basis matrices
Figure BDA0003421528340000061
Sum coefficient matrix
Figure BDA0003421528340000062
Product of (i.e. X ≈ UV)TK < min (m, n). Frobenius norm is adopted to measure original matrix X and UVTThe objective function of NMF can be expressed as follows:
Figure BDA0003421528340000063
wherein | · | purpleFRepresenting the Frobenius norm.
Constructing nearest neighbor maps on the local scatter of the sample can effectively model the local geometry, thus introducing smooth regularization on the underlying (local geometry representing the sample in space)iIs xiIn the low-dimensional representation in the new coding matrix, we can use euclidean distances to measure the smoothness between the low-dimensional representations:
Figure BDA0003421528340000064
where Tr (.) represents the trace of the matrix. L is the Laplace matrix of the graph, L-D-W, D represents a diagonal matrix, and D is the matrixEach term of (1) is WijIs the sum of the rows (or columns, since W is symmetrical), i.e. Dii＝∑jWij. W is a weight matrix, the element W in the matrixijThe definition is as follows:
Figure BDA0003421528340000065
wherein
Figure BDA0003421528340000066
Representing a set of K adjacent data points, X ═ X1,x2,...,xnIs a set of vertices.
The global geometry of the space is represented using a transformation and incorporated into the NMF algorithm as an additional principal component graph regularization term. In particular, global scattering on the coding matrix is maximized and defined as
Figure BDA0003421528340000071
Wherein,
Figure BDA0003421528340000072
representing the average of the encoded vectors.
Equation (4) is further simplified and yields:
Figure BDA0003421528340000073
where M ═ I-E is called principal component diagram, E ═ 1/n) eeTI is an n × n identity matrix and e is an n-dimensional column vector with elements equal to 1.
Applying graph regularization term constraint to the original NMF model through joint modeling (2) and (5), and defining the obtained objective function of non-negative matrix decomposition as:
Figure BDA0003421528340000074
where alpha and beta are two trade-off parameters.
Finally, using LPSmoothness constraints constrain the base matrix to obtain a smooth and more accurate solution:
Figure BDA0003421528340000075
where λ is a non-negative parameter.
In practical applications, it is a troublesome problem to effectively deal with noise or abnormal values. The correlation entropy is used in place of the euclidean norm in the error metric to improve the robustness of the algorithm:
entropy is a measure of the non-linearity and local similarity of two random variables x and y. The definition is as follows:
C(x,y)＝E[k(x,y)] (8)
e [ ] and k (,) are the desired operator and kernel functions, respectively, that satisfy Mercer's theory.
The invention uses the Gaussian kernel function as the kernel function of the correlation entropy to obtain the following formula:
Figure BDA0003421528340000081
where σ is the core bandwidth parameter, σ > 0. If x and y are vectors, then the Gaussian kernel function is:
Figure BDA0003421528340000082
since the joint distribution function of the random variables x and y is usually unknown, the available data samples
Figure BDA0003421528340000083
Is limited. The associated entropy of a sample can be estimated using the following formula:
Figure BDA0003421528340000084
the maximization of formula (11) is called maximum entropy criterion (MCC), which has the advantage that higher order matrices can be considered, and therefore, adding the maximum correlation entropy criterion in non-negative matrix factorization can make the algorithm more robust in handling outliers and noise.
Using corentropy instead of euclidean norm to improve the robustness of the algorithm, the entropy objective function can be written as follows:
Figure BDA0003421528340000085
obviously, the objective function of entropy is non-quadratic and non-convex, and it is difficult to directly optimize the solution. The HQ technique based on the convex conjugate function theory can effectively solve the above optimization problem. It converts the relevant entropy terms in the objective function into quadratic terms in multiplicative form. By using the properties of the convex conjugate function, it is defined as follows:
the convex conjugate function φ () where g (x) exists is such that:
Figure BDA0003421528340000086
wherein if x has been determined, the above formula will reach a maximum at z ═ -g (x).
By definition, substituting equation (13) into the objective function, one can obtain:
Figure BDA0003421528340000087
wherein z is [ z ]1,…,zM]TRepresenting the auxiliary vector. Maximizing the augmented objective function relative to z by fixing U and V yields:
Figure BDA0003421528340000091
the kernel bandwidth parameter σ is updated according to the following iterative equation:
Figure BDA0003421528340000092
the objective function can be rewritten as a maximization problem as follows:
Figure BDA0003421528340000093
for ease of computation, the objective function is equivalently transformed into the following minimization problem:
Figure BDA0003421528340000094
because the robust local and global regularized non-negative matrix factorization clustering method provided by the invention is non-convex and cannot find a global optimal solution, an alternative iteration strategy can be adopted to solve the robust local and global regularized non-negative matrix factorization clustering method model so as to obtain a local optimal solution. Thus, the objective function can be written in the form:
Figure BDA0003421528340000095
wherein Q represents a diagonal matrix with diagonal elements QiiExpressed as:
Figure BDA0003421528340000096
a multiplicative iterative algorithm is used to solve equation f. Constraint U for U and V respectivelyik≥0,vjkNot less than 0, introducing Lagrange coefficient psiikAnd phijkLet psi ═ psiik]，Φ＝[φjk]The lagrange function L can be expressed as:
L＝Tr(XXT)-2Tr(XVUT)+Tr(UVTVUT) +αTr(VLVT)-βTr(VMVT)+2λ||U||P (21) +Tr(ΨUT)+Tr(ΦVT)
the partial derivatives of U and V can be found:
Figure BDA0003421528340000101
from L ═ D-W, M ═ I-E, the updated rule for variable U, V is derived:
the update rule of the variable U is as follows:
Figure BDA0003421528340000102
the update rule of the variable V is as follows:
Figure BDA0003421528340000103
the foregoing illustrates and describes the principles, general features, and advantages of the present invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (7)

1. A robust local and global regularized non-negative matrix factorization clustering method is characterized by comprising the following steps:
step S1: acquiring an image clustering sample;
step S20: constructing a nearest neighbor map on the local scatter of the sample obtained in the step S1 and introducing smooth regularization;
step S30: representing the global geometry of the space by using transformation, and taking the global geometry as an additional main component graph regularization item to be incorporated into an NMF algorithm;
step S40: applying graph regularization term constraints to an original NMF model by joint modeling and utilizing LPThe smoothness constraint restrains the base matrix;
step S50: using correlation entropy to replace Euclidean norm in the error measurement, thereby obtaining a robust target function of local and global regularization nonnegative matrix decomposition;
step S60: iterating preset times by using an iterative weighting method according to the target function, updating the variable U, V, and completing robust local and global regularized nonnegative matrix decomposition;
step S70: and (5) carrying out clustering analysis on the coefficient matrix by adopting a K-means clustering algorithm.
2. The robust local and global regularized non-negative matrix factorization clustering method according to claim 1, wherein in the step S20, the smooth regularization is specifically:
Figure FDA0003421528330000011
where Tr (·) denotes the trace of the matrix, L is the laplacian matrix of the graph, L-D-W, D denotes a diagonal matrix, and each entry in D is WijOf rows (a) or (b), i.e. Dii＝∑jWijW is a weight matrix, the element W in the matrixijThe definition is as follows:
Figure FDA0003421528330000021
wherein
Figure FDA0003421528330000027
Representing a set of K adjacent data points, X ═ X1,x2,...,xnIs a set of vertices.
3. A robust local and global regularized non-negative matrix factorization clustering method as claimed in claim 2, wherein in said step S30, the global scattering on the coding matrix is maximized and defined as:
Figure FDA0003421528330000022
wherein,
Figure FDA0003421528330000023
represents an average value of the encoded vector;
the equation is further simplified and yields:
Figure FDA0003421528330000024
where M ═ I-E is called principal component diagram, E ═ 1/n) eeTI is an n × n identity matrix and e is an n-dimensional column vector with elements equal to 1.
4. The robust local and global regularized non-negative matrix factorization clustering method according to claim 3, wherein in the step S40, the obtained objective function of the non-negative matrix factorization is specifically:
Figure FDA0003421528330000025
where α and β are two trade-off parameters;
by means of LPThe smoothness constraint constrains the base matrix to obtain a smooth and more accurate solution, specifically:
Figure FDA0003421528330000026
where λ is a non-negative parameter.
5. The robust local and global regularized non-negative matrix factorization clustering method according to claim 4, wherein in the step S50, an objective function of the robust local and global regularized non-negative matrix factorization is specifically:
Figure FDA0003421528330000031
the first term of the equation is based on entropy reconstruction error, the second term is a local smoothness graph regular term, the third term represents a global geometry structure graph regular term, and the fourth term utilizes LPSmoothness constraints constrain the base matrix.
6. The robust local and global regularized non-negative matrix factorization clustering method according to claim 5, wherein in said step S60, comprising:
establishing a Lagrangian function L according to the objective function of the non-negative matrix factorization:
L＝Tr(XXT)-2Tr(XVUT)+Tr(UVTVUT)+αTr(VLVT)-βTr(VMVT)+2λ||U||P+Tr(ΨUT)+Tr(ΦVT)
wherein Ψ ═ ψik]，Φ＝[φjk]；
Respectively solving the partial derivatives of the basic matrix U and the coefficient matrix V by using the Karush-Kuhn-Tucker condition
Figure FDA0003421528330000034
φjkνjkObtaining the iterative expressions of the basic matrix U and the coefficient matrix V respectively as 0:
the update rule of the variable U is as follows:
Figure FDA0003421528330000032
the update rule of the variable V is as follows:
Figure FDA0003421528330000033
7. the robust local and global regularized non-negative matrix factorization clustering method according to claim 6, wherein in the step S70, the matrix U and the matrix V are separately subjected to partial derivation, and Karush-Kuhn-Tucker condition is used
Figure FDA0003421528330000035
φjkνjkObtaining an iterative equation of the basis matrix U and the coefficient matrix V as 0, further including:
performing loop iteration according to the initialization weight, the basic matrix U and the iterative expression of the coefficient matrix V;
and after the cycle reaches the preset iteration times t, outputting a basic matrix U and a coefficient matrix V to finish the robust local and global regularized nonnegative matrix decomposition.
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CN115577564A (en) * 2022-11-11 2023-01-06 江西师范大学 Robust nonnegative matrix decomposition method and system for multi-constraint adaptive graph learning
CN115810108A (en) * 2022-09-23 2023-03-17 南京审计大学 Image feature extraction method in big data audit based on REGNMF
CN118197342A (en) * 2024-05-15 2024-06-14 电子科技大学中山学院 High-precision audio signal denoising method based on improved NMF and K-means++

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CN115810108A (en) * 2022-09-23 2023-03-17 南京审计大学 Image feature extraction method in big data audit based on REGNMF
CN115810108B (en) * 2022-09-23 2023-08-08 南京审计大学 Image feature extraction method in big data audit based on REGNMF
CN115577564A (en) * 2022-11-11 2023-01-06 江西师范大学 Robust nonnegative matrix decomposition method and system for multi-constraint adaptive graph learning
CN115577564B (en) * 2022-11-11 2023-08-22 江西师范大学 Robust non-negative matrix factorization method and system for multi-constraint adaptive graph learning
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