# Combinatorial Regularity for Relatively Perfect Discrete Morse Gradient Vector Fields of ReLU Neural Networks

arXiv: https://arxiv.org/abs/2412.18005 · HTML: https://mlsanity.com/paper/2412.18005/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 32 references that are in the map.
- First posted on arXiv 23 Dec 2024.

## Builds on (most cited first, 9 of 9)

- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [The Geometry of Deep Networks: Power Diagram Subdivision](https://mlsanity.com/paper/1905.08443/) (2019)
- [On Characterizing the Capacity of Neural Networks using Algebraic Topology](https://mlsanity.com/paper/1802.04443/) (2018)
- [On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks](https://mlsanity.com/paper/2008.09052/) (2020)
- [Provable Certificates for Adversarial Examples: Fitting a Ball in the Union of Polytopes](https://mlsanity.com/paper/1903.08778/) (2019)
- [Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks](https://mlsanity.com/paper/2207.07696/) (2022)
- [Topological Expressivity of ReLU Neural Networks](https://mlsanity.com/paper/2310.11130/) (2023)
- [Local and global topological complexity measures OF ReLU neural network functions](https://mlsanity.com/paper/2204.06062/) (2022)
- [Critical Sets of PL and Discrete Morse Theory: a Correspondence](https://mlsanity.com/paper/2001.04767/) (2020)

## Abstract (arXiv)

One common function class in machine learning is the class of ReLU neural networks. ReLU neural networks induce a piecewise linear decomposition of their input space called the canonical polyhedral complex. It has previously been established that it is decidable whether a ReLU neural network is piecewise linear Morse. In order to expand computational tools for analyzing the topological properties of ReLU neural networks, and to harness the strengths of discrete Morse theory, we introduce a schematic for translating between a given piecewise linear Morse function (e.g. parameters of a ReLU neural network) on a canonical polyhedral complex and a compatible (``relatively perfect") discrete Morse function on the same complex. Our approach is constructive, producing an algorithm that can be used to determine if a given vertex in a canonical polyhedral complex corresponds to a piecewise linear Morse critical point. Furthermore we provide an algorithm for constructing a consistent discrete Morse pairing on cells in the canonical polyhedral complex which contain this vertex. We additionally provide some new realizability results with respect to sublevel set topology in the case of shallow ReLU neural networks.