On Characterizing the Capacity of Neural Networks using Algebraic Topology

TL;DR

This paper explores how algebraic topology can measure data complexity and influence neural network capacity, revealing topological phase transitions that affect generalization and architecture selection.

Contribution

It introduces algebraic topology as a novel measure for data complexity and empirically characterizes neural networks' topological capacity and phase transitions.

Findings

Neural networks exhibit topological phase transitions at various dataset complexities.

Topological complexity limits the expressive power and generalization of neural networks.

The study connects topological measures to neural architecture choices.

Abstract

The learnability of different neural architectures can be characterized directly by computable measures of data complexity. In this paper, we reframe the problem of architecture selection as understanding how data determines the most expressive and generalizable architectures suited to that data, beyond inductive bias. After suggesting algebraic topology as a measure for data complexity, we show that the power of a network to express the topological complexity of a dataset in its decision region is a strictly limiting factor in its ability to generalize. We then provide the first empirical characterization of the topological capacity of neural networks. Our empirical analysis shows that at every level of dataset complexity, neural networks exhibit topological phase transitions. This observation allowed us to connect existing theory to empirically driven conjectures on the choice of architectures for fully-connected neural networks.