The Geometry of Machine Learning Models

TL;DR

This paper introduces a geometric framework for analyzing machine learning models using Riemannian simplicial complexes, enabling new insights into model structure, regularization, and data-induced curvature effects.

Contribution

It develops a novel geometric approach to model analysis, including geometric regularization and curvature measures, advancing understanding of model structure and learning dynamics.

Findings

Framework captures adjacency and geometric properties of model partitions.

Introduces geometric regularization techniques for model refinement.

Defines discrete curvature measures to analyze data distribution effects.

Abstract

This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships but also geometric properties including cell volumes, volumes of faces where cells meet, and dihedral angles between adjacent cells. For neural networks, we introduce a differential forms approach that tracks geometric structure through layers via pullback operations, making computations tractable by focusing on data-containing cells. The framework enables geometric regularization that directly penalizes problematic spatial configurations and provides new tools for model refinement through extended Laplacians and simplicial splines. We also explore how data distribution induces effective geometric curvature in model partitions, developing discrete curvature measures for vertices that quantify local geometric complexity and statistical Ricci curvature for edges that captures pairwise relationships between cells. While focused on mathematical foundations, this geometric perspective offers new approaches to model interpretation, regularization, and diagnostic tools for understanding learning dynamics.