The Geometry of Machine Learning Models

2025-08-04

AI Summary

This paper addresses the lack of geometric characterization for the implicit spatial partitioning induced by machine learning models. We propose a modeling framework based on Riemannian simplicial complexes: model decision regions are represented as metric-bearing simplicial complexes, enabling systematic quantification of geometric features—including volumes, facet areas, and dihedral angles. To track geometric evolution across neural network layers, we introduce pullbacks of differential forms and an extended Laplacian operator. Furthermore, we define vertex-wise discrete curvature and edge-wise statistical Ricci curvature to explicitly link model geometry with underlying data distributions. The resulting geometric regularization method directly constrains spatial configurations, enhancing both generalization and interpretability. Empirically and theoretically, it demonstrates consistency in regularizer design and efficacy in diagnosing learning dynamics, offering a computationally tractable and principled approach to geometric deep learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Analyzing machine learning models via geometric partition properties
• Introducing geometric regularization to penalize spatial configurations
• Developing discrete curvature measures for model interpretation

Innovation

• Riemannian simplicial complexes for model partitions
• Differential forms approach via pullback operations
• Discrete curvature measures for geometric complexity