Systems and methods for robust low-rank matrix approximation
원문보기
IPC분류정보
국가/구분
United States(US) Patent
등록
국제특허분류(IPC7판)
H04N-019/10
G06F-017/17
G06F-007/02
G06F-007/76
출원번호
US-0676600
(2017-08-14)
등록번호
US-10229092
(2019-03-12)
발명자
/ 주소
Zeng, Wen-Jun
So, Hing Cheung
Chen, Jiayi
출원인 / 주소
City University of Hong Kong
대리인 / 주소
Norton Rose Fulbright US LLP
인용정보
피인용 횟수 :
0인용 특허 :
236
초록▼
Systems and methods which provide robust low-rank matrix approximation using low-rank matrix factorization in the lp-norm space, where p2 (e.g., 1≤p2), providing a lp-PCA technique are described. For example, embodiments are configured to provide robust low-rank matrix approximation using low-rank m
Systems and methods which provide robust low-rank matrix approximation using low-rank matrix factorization in the lp-norm space, where p2 (e.g., 1≤p2), providing a lp-PCA technique are described. For example, embodiments are configured to provide robust low-rank matrix approximation using low-rank matrix factorization in the least absolute deviation (l1-norm) space providing a l1-PCA technique. Embodiments minimize the lp-norm of the residual matrix in the subspace factorization of an observed data matrix, such as to minimize the l1-norm of the residual matrix where p=1. The alternating direction method of multipliers (ADMM) is applied according to embodiments to solve the subspace decomposition of the low-rank matrix factorization with respect to the observed data matrix. Iterations of the ADMM may comprise solving a l2-subspace decomposition and calculating the proximity operator of the l1-norm.
대표청구항▼
1. A method for low-rank approximation of an observed data matrix, the method comprising: obtaining, by a processor-based system, the observed data matrix;performing, by logic of the processor-based system, factorization of the observed data matrix in lp-norm space, wherein p<2; andproviding, by the
1. A method for low-rank approximation of an observed data matrix, the method comprising: obtaining, by a processor-based system, the observed data matrix;performing, by logic of the processor-based system, factorization of the observed data matrix in lp-norm space, wherein p<2; andproviding, by the processor-based system from a result of the lp-norm space factorization of the observed data matrix, a low-rank approximation comprising principal components extracted from the observed data matrix. 2. The method of claim 1, wherein obtaining the observed data matrix comprises: deriving the observed data matrix from a signal selected from the group consisting of a data signal, a graphical signal, an audio signal, a video signal, and a multimedia signal, wherein the observed data matrix comprises a matrix of a plurality of data points representing the signal. 3. The method of claim 2, wherein providing the low-rank approximation comprises: providing the low-rank approximation for use at least one application selected from the group consisting of a surveillance application, a machine learning application, a web search application, a bioinformatics application, a dimensionality reduction application, and a signal processing application. 4. The method of claim 1, wherein the observed data matrix comprises presence of at least one of impulsive noise, outliers, or sparse features. 5. The method of claim 1, wherein 1≤p<2. 6. The method of claim 5, wherein performing the factorization of the observed data matrix in the lp-norm space comprises: performing factorization of the observed data matrix in l1-norm space, wherein p=1. 7. The method of claim 1, wherein performing the factorization of the observed data matrix in the lp-norm space comprises: applying alternating direction method of multipliers (ADMM) to solve subspace decomposition of low-rank matrix factorization with respect to the observed data matrix. 8. The method of claim 7, wherein applying the ADMM to solve subspace decomposition of low-rank matrix factorization comprises: minimizing a residual matrix in a subspace factorization of the observed data matrix. 9. The method of claim 7, wherein applying the ADMM to solve subspace decomposition of low-rank matrix factorization comprises: solving a l2-subspace decomposition; andcalculating a proximity operator of the lp-norm. 10. The method of claim 9, wherein solving the l2-subspace decomposition comprises: using least Frobenius norm solved by truncated singular value decomposition (SVD). 11. The method of claim 9, wherein calculating the proximity operator of the lp,-norm comprises: using a closed-form soft-thresholding operator for complex variables. 12. The method of claim 9, wherein solving the l2-subspace decomposition and calculating to proximity operator of the lp-norm are performed in each iterative step of the ADMM. 13. A system for low-rank approximation of an observed data matrix, the system comprising: one or more data processors; andone or more non-transitory computer-readable storage media containing program code configured to cause the one or more data processors to perform operations including: obtain the observed data matrix;perform factorization of the observed data matrix in lp-norm space, wherein p<2; andprovide, from a result of the lp-norm space factorization of the observed data matrix, a low-rank approximation comprising principal components extracted from the observed data matrix. 14. The system of claim 13, wherein the program code configured to cause the one or more data processors to obtain the observed data matrix further causes the one or more data processors to: derive the observed data matrix from a signal selected from the group consisting of a data signal, a graphical signal, an audio signal, a video signal, and a multimedia signal, wherein the observed data matrix comprises a matrix of a plurality of data points representing the signal. 15. The system of claim 14, wherein the program code configured to cause the one or more data processors to provide the low-rank approximation further causes the one or more data processors to: provide the low-rank approximation for use at least one application selected from the group consisting of a surveillance application, a machine learning application, a web search application, a bioinformatics application, a dimensionality reduction application, and a signal processing application. 16. The system of claim 13, wherein the observed data matrix comprises presence of at least one of impulsive noise, outliers, or sparse features. 17. The system of claim 13, wherein the program code configured to cause the one or more data processors to perform the factorization of the observed data matrix in the lp-norm space further causes the one or more data processors to: perform factorization of the observed data matrix in l1-norm space, wherein p=1. 18. The system of claim 13, wherein the program code configured to cause the one or more data processors to perform the factorization of the observed data matrix in the lp-norm space further causes the one or more data processors to: apply alternating direction method of multipliers (ADMM) to solve subspace decomposition of low-rank matrix factorization with respect to the observed data matrix. 19. The system of claim 18, wherein the program code configured to cause the one or more data processors to apply the ADMM to solve subspace decomposition of low-rank matrix factorization further causes the one or more data processors to: minimize a residual matrix in a subspace factorization of the observed data matrix. 20. The system of claim 18, wherein the program code configured to cause the one or more data processors to apply the ADMM to solve subspace decomposition of low-rank matrix factorization further causes the one or more data processors to: solve a l2-subspace decomposition; andcalculate a proximity operator of the lp-norm. 21. The system of claim 20, wherein the program code configured to cause the one or more data processors to solve the l2-subspace decomposition further causes the one or more data processors to: use least Frobenius norm solved by truncated singular value decomposition (SVD). 22. The system of claim 20, wherein the program code configured to cause the one or more data processors to calculate the proximity operator of the lp-norm further causes the one or more data processors to: use a closed-form soft-thresholding operator for complex variables. 23. A method for low-rank approximation of an observed data matrix, the method comprising: obtaining, by a processor-based system, the observed data matrix;performing, by logic of the processor-based system, factorization of the observed data matrix in l1-norm space by applying alternating direction method of multipliers (ADMM) to solve subspace decomposition of low-rank matrix factorization with respect to the observed data matrix; andproviding, by the processor-based system from a result of the l1-norm space factorization of the observed data matrix, a low-rank approximation comprising principal components extracted from the observed data matrix. 24. The method of claim 23, wherein applying the ADMM to solve subspace decomposition of low-rank matrix factorization comprises: minimizing a residual matrix in a subspace factorization of the observed data matrix. 25. The method of claim 23, wherein applying the ADMM to solve subspace decomposition of low-rank matrix factorization comprises: performing iterative steps of the ADMM, wherein each iterative step of the ADMM includes: solving a l2-subspace decomposition using least Frobenius norm solved by truncated singular value decomposition (SVD); andcalculating a proximity operator of the lp-norm using a closed-form soft-thresholding operator for complex variables.
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