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

2025-05-31

AI Summary

This paper addresses the fundamental question of whether neural networks can universally approximate matrix inversion, revealing inherent theoretical limitations. Method: Extending the Lipschitz function analysis framework, we rigorously prove— for the first time—that no neural network architecture can universally approximate the inverse of arbitrary invertible matrices. Leveraging spectral modeling of matrices, we derive a precise, necessary and sufficient criterion for efficient neural matrix inversion: bounded condition number and controlled eigenvalue distribution. Our approach integrates generalized Lipschitz-theoretic analysis, spectral-structure modeling, and empirical validation on diverse synthetic and real-world matrix datasets. Results: Under the derived criterion, neural methods achieve high-accuracy, parallelizable matrix inversion; conversely, when violated, approximation error grows exponentially with matrix ill-conditioning. This work establishes rigorous theoretical boundaries and practical guidelines for deploying neural networks to accelerate matrix computations in scientific computing.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science

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.

Problem

• Theoretical limits of neural networks for general matrix inversion
• Conditions enabling neural networks to approximate matrix inverses
• Experimental validation of neural networks on matrix inversion tasks

Innovation

• Expands Lipschitz functions for neural models
• Identifies conditions for matrix inversion approximation
• Tests neural networks on diverse matrix datasets