自动化计算机器学习中随机矩阵的迹，提升理论分析效率。
auto-fpt: Automating Free Probability Theory Calculations for Machine Learning Theory
- 基于自由概率论，自动推导固定点方程系统。
- 可高效求解高维神经网络误差等复杂问题。
- 适合从事理论机器学习研究的学者使用。
现代机器学习理论的许多核心问题涉及大型矩形随机矩阵有理表达式的高维期望迹计算。为通过自由概率论符号化求解此类量，我们提出 auto-fpt——一个轻量级的 Python 与 SymPy 工具，能自动生成可求解的目标量的简化固定点方程组，实质构成一套理论框架。本文概述了 auto-fpt 的算法思想及其在多个有趣问题中的应用，如线性化前馈神经网络的高维误差，成功复现了已有经典结果。我们希望 auto-fpt 能简化高维分析中的大部分计算，助力机器学习社区复现已知现象并发现新规律。
原文摘要 · Abstract (English)
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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