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arXiv · 2601.19250

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

Weiwei Xu·Weijie Shen·Zhengjian Bai·Chen Xu
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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.

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Weiwei Xu, Weijie Shen, Zhengjian Bai, Chen Xu. 2026-01-27. Precision-induced Adaptive Randomized Low-Rank Approximation for SVD and Matrix Inversion. https://arxiv.org/abs/2601.19250

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