Smoothing of adaptive eigenvector extraction in nested orthogonal complement structure with minimum disturbance principle

Abstract

For adaptive extraction of generalized eigensubspace, Nguyen, Takahashi and Yamada proposed a scheme for solving generalized Hermitian eigenvalue problem based on nested orthogonal complement structure. This scheme can extract multiple generalized eigenvectors by combining with any algorithm designed for estimation of the first minor generalized eigenvector. In this paper, we carefully analyse the effect of a discontinuous function employed in the scheme, and show that the discontinuous function can cause unsmooth changes of the estimates by the scheme in its adaptive implementation. To remedy the weakness, we newly introduce a projection step, for smoothing, without increasing the order of the computational complexity. Numerical experiments show that the learning curves of the non-first generalized eigenvectors are improved drastically through the proposed smoothing even when the original scheme results in unexpected performance degradation.