Improved eigenvalue shrinkage using weighted Chebyshev polynomial approximation

Abstract

We propose an eigenvalue shrinkage method with a modified Chebyshev polynomial approximation (CPA). The eigenvalue shrinkage has been used in many fields of signal and image processing. However, the shrinkage takes enormous computation time especially in the case that a matrix constructed from a signal or image becomes very large, i.e., eigendecomposition can hardly be performed. The CPA is an approximation method of the shrinkage function that avoids the eigendecomposition of the matrix. Unfortunately, it is known that the CPA generates Gibbs phenomenon around points of discontinuity for approximating an ideal response. The Chebyshev-Jackson polynomial approximation (CJPA) will alleviate the problem, but the transition bandwidth becomes wide, which is an undesired characteristic for some applications. In this paper, we propose an eigenvalue shrinkage method with the reduced Gibbs phenomenon by modifying the CPA using the weighted least squares approach. Our method can reduce the error as well as the CJPA. Furthermore, it yields the narrow transition band. Some experimental results on spectral clustering validate the effectiveness of the method.