Precision-induced Adaptive Randomized Low-Rank Approximation for SVD and Matrix Inversion

TL;DR

This paper introduces a novel adaptive randomized approach for low-rank approximation in SVD and matrix inversion, which automatically determines the optimal rank based on singular value distribution, reducing computational costs.

Contribution

It proposes a precision-guided re-normalization method that eliminates the need for rank guessing, enabling efficient and accurate approximations for large matrices.

Findings

Algorithms achieve accurate approximations at reduced computational cost.

Theoretical analysis supports the effectiveness of the approach.

Numerical experiments demonstrate superior performance over existing methods.

Abstract

Singular value decomposition (SVD) and matrix inversion are ubiquitous in scientific computing. Both tasks are computationally demanding for large scale matrices. Existing algorithms can approximatively solve these problems with a given rank, which however is unknown in practice and requires considerable cost for tuning. In this paper, we tackle the SVD and matrix inversion problems from a new angle, where the optimal rank for the approximate solution is explicitly guided by the distribution of the singular values. Under the framework, we propose a precision-induced random re-normalization procedure for the considered problems without the need of guessing a good rank. The new algorithms built upon the procedure simultaneously calculate the optimal rank for the task at a desired precision level and lead to the corresponding approximate solution with a substantially reduced computational cost. The promising performance of the new algorithms is supported by both theory and numerical examples.