  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.
