A Theory of Diversity for Random Matrices with Applications to In-Context Learning of Schr"odinger Equations
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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.
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References · 11
- 2017Attention Is All You Needcited by 60,361
- 2022What Can Transformers Learn In-Context? A Case Study of Simple Function Classescited by 715
- 2023Trained Transformers Learn Linear Models In-Contextcited by 337
- 2023Transformers learn to implement preconditioned gradient descent for in-context learningcited by 279
- 2023One Step of Gradient Descent is Provably the Optimal In-Context Learner with One Layer of Linear Self-Attentioncited by 170
- 2023In-Context Operator Learning with Data Prompts for Differential Equation Problemscited by 101
- 2024Asymptotic theory of in-context learning by linear attentioncited by 55
- 2024Continuum Attention for Neural Operatorscited by 40
- 2024Zebra: In-Context Generative Pretraining for Solving Parametric PDEscited by 16
- 2024In-Context Learning of Linear Systems: Generalization Theory and Applications to Operator Learningcited by 6
- 2025Continuum Transformers Perform In-Context Learning by Operator Gradient Descentcited by 3
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