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

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.

@article{subramonian2025_2504.10754,
  title={ auto-fpt: Automating Free Probability Theory Calculations for Machine Learning Theory },
  author={ Arjun Subramonian and Elvis Dohmatob },
  journal={arXiv preprint arXiv:2504.10754},
  year={ 2025 }
}