This talk is intended as an introduction to and overview of the geometric topology of (some) artificial neural network functions, with aims towards advancing the understanding of deep learning models. First, I will introduce the type of neural network functions under consideration (“ReLU networks,” or multilayer perceptrons with ReLU activation) and their relation to contemporary machine learning.

I will then discuss some of the perspectives from which geometric and topological measures of ReLU networks may be exploited to understand and analyze the structure of individual such functions as well as the class of all such functions, both theoretically and computationally. I will pay special attention towards the “decision region/boundary” interpretation of classification models, which corresponds to (sub)level set approximation. A concern is that sublevel sets with too high of complexity (as measured via topological invariants) corresponds to “memorization/overfitting” of a machine learning model, but sublevel sets of insufficient complexity will fail to generalize over a large portion of the problem domain. Resultingly, we seek tools to assess the complexity of a given ReLU network.

One result in this direction is that, under certain genericity and transversality assumptions on intermediate layers, the (mod-2) Betti numbers of level sets of ReLU networks can be computed exactly by exploiting hyperplane arrangement combinatorics.

I will additionally discuss some of the unique challenges faced when extending piecewise linear and discrete Morse theory to this function class, including current progress.

This talk is based on, in part, work done jointly with J. Elisenda Grigsby, Kathryn Lindsey, and Robyn Brooks.

This talk is based on, in part, work done jointly with J. Elisenda Grigsby, Kathryn Lindsey, and Robyn Brooks. References include:

Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks. Marissa Masden, 2022. (To appear, SIAM Journal of Applied Algebra and Geometry). https://arxiv.org/abs/2207.07696

Local and global topological complexity measures of ReLU neural network functions. J. Elisenda Grigsby, Kathryn Lindsey and Marissa Masden, 2022. https://arxiv.org/abs/2204.06062

Combinatorial Regularity for Relatively Perfect Discrete Morse Gradient Vector Fields on ReLU Neural Networks. Robyn Brooks, Marissa Masden, 2024. https://arxiv.org/abs/2412.18005