Fast Singular Value Shrinkage with Chebyshev Polynomial Approximation Based on Signal Sparsity
This work addresses computational bottlenecks in matrix rank minimization for high-dimensional signals like images, offering a faster alternative for signal processing practitioners, though it is incremental as it builds on existing thresholding methods with CPA.
The paper tackles the high computational cost of singular value decomposition (SVD) in matrix rank minimization for signal processing by proposing a Chebyshev polynomial approximation (CPA) method to approximate singular value thresholding without computing singular values and vectors, significantly reducing computation time while maintaining approximation precision in image processing applications.
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.