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

arXiv: https://arxiv.org/abs/2601.12587 · HTML: https://mlsanity.com/paper/2601.12587/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 20 references that are in the map.
- First posted on arXiv 18 Jan 2026.

## Builds on (most cited first, 11 of 11)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [Trained Transformers Learn Linear Models In-Context](https://mlsanity.com/paper/2306.09927/) (2023)
- [Transformers learn to implement preconditioned gradient descent for in-context learning](https://mlsanity.com/paper/2306.00297/) (2023)
- [One Step of Gradient Descent is Provably the Optimal In-Context Learner with One Layer of Linear Self-Attention](https://mlsanity.com/paper/2307.03576/) (2023)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Asymptotic theory of in-context learning by linear attention](https://mlsanity.com/paper/2405.11751/) (2024)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [Zebra: In-Context Generative Pretraining for Solving Parametric PDEs](https://mlsanity.com/paper/2410.03437/) (2024)
- [In-Context Learning of Linear Systems: Generalization Theory and Applications to Operator Learning](https://mlsanity.com/paper/2409.12293/) (2024)
- [Continuum Transformers Perform In-Context Learning by Operator Gradient Descent](https://mlsanity.com/paper/2505.17838/) (2025)

## Abstract (arXiv)

We address the following question: given a collection A(1), dots, A(N) of independent d × d random matrices drawn from a common distribution mathbbP, what is the probability that the centralizer of A(1), dots, 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.