auto-fpt: Automating Free Probability Theory Calculations for Machine Learning Theory

Arjun Subramonian; Elvis Dohmatob

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arXiv · 2504.10754

auto-fpt: Automating Free Probability Theory Calculations for Machine Learning Theory

Arjun Subramonian·Elvis Dohmatob

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Abstract

A large part of modern machine learning theory often involves computing the high-dimensional expected trace of a rational expression of large rectangular random matrices. To symbolically compute such quantities using free probability theory, we introduce auto-fpt, a lightweight Python and SymPy-based tool that can automatically produce a reduced system of fixed-point equations which can be solved for the quantities of interest, and effectively constitutes a theory. We overview the algorithmic ideas underlying auto-fpt and its applications to various interesting problems, such as the high-dimensional error of linearized feed-forward neural networks, recovering well-known results. We hope that auto-fpt streamlines the majority of calculations involved in high-dimensional analysis, while helping the machine learning community reproduce known and uncover new phenomena.

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Arjun Subramonian, Elvis Dohmatob. 2025-04-14. auto-fpt: Automating Free Probability Theory Calculations for Machine Learning Theory. https://arxiv.org/abs/2504.10754

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