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

arXiv: https://arxiv.org/abs/2506.00642 · HTML: https://mlsanity.com/paper/2506.00642/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 33 references that are in the map.
- First posted on arXiv 31 May 2025.

## Builds on (most cited first, 8 of 8)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [The Lipschitz Constant of Self-Attention](https://mlsanity.com/paper/2006.04710/) (2020)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows](https://mlsanity.com/paper/2206.02102/) (2022)

## Built on by (most cited first, 1 of 1)

- [NeuMatC: A General Neural Framework for Fast Parametric Matrix Operation](https://mlsanity.com/paper/2511.22934/) (2025)

## Abstract (arXiv)

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.