Singular Value Decomposition Approximation via Kronecker Summations for Imaging Applications

TL;DR

This paper introduces a novel method for approximating the truncated singular value decomposition of large structured matrices using Kronecker summations, improving efficiency in imaging applications.

Contribution

It presents a new approach that decomposes matrices into Kronecker sums to efficiently approximate singular values and vectors, outperforming existing randomized and iterative methods.

Findings

The method accurately approximates singular values and vectors.

Numerical experiments demonstrate improved efficiency.

Application to large-scale inverse problems shows practical benefits.

Abstract

In this paper we propose an approach to approximate a truncated singular value decomposition of a large structured matrix. By first decomposing the matrix into a sum of Kronecker products, our approach can be used to approximate a large number of singular values and vectors more efficiently than other well known schemes, such as randomized matrix algorithms or iterative algorithms based on Golub-Kahan bidiagonalization. We provide theoretical results and numerical experiments to demonstrate the accuracy of our approximation and show how the approximation can be used to solve large scale ill-posed inverse problems, either as an approximate filtering method, or as a preconditioner to accelerate iterative algorithms.