Fast Singular Value Shrinkage with Chebyshev Polynomial Approximation Based on Signal Sparsity
May 19, 2017 ยท Declared Dead ยท ๐ IEEE Transactions on Signal Processing
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Authors
Masaki Onuki, Shunsuke Ono, Keiichiro Shirai, Yuichi Tanaka
arXiv ID
1705.07112
Category
math.NA: Numerical Analysis
Cross-listed
cs.LG
Citations
10
Venue
IEEE Transactions on Signal Processing
Last Checked
1 month ago
Abstract
We propose an approximation method for thresholding of singular values using Chebyshev polynomial approximation (CPA). Many signal processing problems require iterative application of singular value decomposition (SVD) for minimizing the rank of a given data matrix with other cost functions and/or constraints, which is called matrix rank minimization. In matrix rank minimization, singular values of a matrix are shrunk by hard-thresholding, soft-thresholding, or weighted soft-thresholding. However, the computational cost of SVD is generally too expensive to handle high dimensional signals such as images; hence, in this case, matrix rank minimization requires enormous computation time. In this paper, we leverage CPA to (approximately) manipulate singular values without computing singular values and vectors. The thresholding of singular values is expressed by a multiplication of certain matrices, which is derived from a characteristic of CPA. The multiplication is also efficiently computed using the sparsity of signals. As a result, the computational cost is significantly reduced. Experimental results suggest the effectiveness of our method through several image processing applications based on matrix rank minimization with nuclear norm relaxation in terms of computation time and approximation precision.
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