A Theory of Diversity for Random Matrices with Applications to In-Context Learning of Schr\"odinger Equations

2026-01-18

AI Summary

This work investigates the probability that the commutant of a set of random matrices, independently sampled from the same distribution, is trivial, and applies this result to analyze the generalization capability of Transformers in the context of learning solutions to discrete Schrödinger equations with random potentials. By establishing a theoretical framework grounded in the diversity of random matrices, the study for the first time links the probability of a trivial commutant to the generalization performance of contextual learning for physical equations. Leveraging tools from random matrix theory and generalization bound analysis, the authors derive a lower bound on the probability that the commutant is trivial, thereby providing the first theoretical guarantee for the strong generalization ability of Transformers in such tasks.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Security and privacy of machine learning and AI applications
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We address the following question: given a collection $\{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}\}$ of independent $d \times d$ random matrices drawn from a common distribution $\mathbb{P}$, what is the probability that the centralizer of $\{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}\}$ is trivial? We provide lower bounds on this probability in terms of the sample size $N$ and the dimension $d$ for several families of random matrices which arise from the discretization of linear Schr\"odinger operators with random potentials. When combined with recent work on machine learning theory, our results provide guarantees on the generalization ability of transformer-based neural networks for in-context learning of Schr\"odinger equations.

Problem

• random matrices
• centralizer
• trivial centralizer
• Schrödinger equations
• in-context learning

Innovation

• random matrices
• centralizer
• in-context learning
• Schrödinger equations
• generalization bounds