Rethinking Neural-based Matrix Inversion: Why can't, and Where can

TL;DR

This paper investigates the theoretical limitations and potential conditions under which neural networks can approximate matrix inverses, supported by experiments across diverse datasets, highlighting both challenges and opportunities.

Contribution

It provides a theoretical analysis of neural network capabilities and limitations in matrix inversion, expanding Lipschitz function classes and identifying effective approximation conditions.

Findings

Neural networks face fundamental limitations in universal matrix inversion.

Certain conditions enable neural networks to effectively approximate inverses.

Experimental results validate theoretical insights across diverse datasets.

Abstract

Deep neural networks have achieved substantial success across various scientific computing tasks. A pivotal challenge within this domain is the rapid and parallel approximation of matrix inverses, critical for numerous applications. Despite significant progress, there currently exists no universal neural-based method for approximating matrix inversion. This paper presents a theoretical analysis demonstrating the fundamental limitations of neural networks in developing a general matrix inversion model. We expand the class of Lipschitz functions to encompass a wider array of neural network models, thereby refining our theoretical approach. Moreover, we delineate specific conditions under which neural networks can effectively approximate matrix inverses. Our theoretical results are supported by experimental results from diverse matrix datasets, exploring the efficacy of neural networks in addressing the matrix inversion challenge.