Exact deflation for accurate SVD computation of nonnegative bidiagonal products of arbitrary rank
Rong Huang, Jungong Xue
TL;DR
This work addresses accurate SVD computation for nonnegative bidiagonal products of arbitrary rank, including rank-deficient and rectangular factors. It introduces an exact deflation framework combined with an extracting method for arbitrary submatrices, enabling exact zero-singular-value deflation and high relative accuracy for the nonzero spectrum. A periodic deflation strategy reduces the dominant cost to $O(rS+\max\{n_{0}^{2}r,n_{K}^{2}r\})$, while the nonzero singular values are obtained via a standard full-rank bidiagonal SVD on $\bar B$. The approach extends to wide classes of structured TN matrices and arbitrary submatrices, with validation in numerical experiments showing robust performance under ill-conditioning and rank deficiency.
Abstract
Dealing with zero singular values can be quite challenging, as they have the potential to cause numerous numerical difficulties. This paper presents a method for computing the singular value decomposition (SVD) of a nonnegative bidiagonal product of arbitrary rank, regardless of whether the factors are of full rank or rank-deficient, square or rectangular. A key feature of our method is its ability to exactly deflate all zero singular values with a favorable complexity, irrespective of rank deficiency and ill conditioning. Furthermore, it ensures the computation of nonzero singular values, no matter how small they may be, with high relative accuracy. Additionally, our method is well-suited for accurately computing the SVDs of arbitrary submatrices, leveraging an approach to extract their representations from the original product. We have conducted error analysis and numerical experiments to validate the claimed high relative accuracy.