On Characterizing the Capacity of Neural Networks using Algebraic Topology
This work addresses the challenge of choosing neural architectures based on data properties, offering a theoretical and empirical framework that could guide design decisions in machine learning.
The paper tackles the problem of architecture selection by linking data complexity, measured via algebraic topology, to neural network expressivity and generalization, showing that networks exhibit topological phase transitions across dataset complexities.
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.