# Constructive Bounds for the Topological Expressivity of Neural Networks

arXiv: https://arxiv.org/abs/2610.13931 · HTML: https://mlsanity.com/paper/2610.13931/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 37 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Builds on (most cited first, 23 of 23)

- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Implicit Diffusion Models for Continuous Super-Resolution](https://mlsanity.com/paper/2303.16491/) (2023)
- [Deep Learning-Based Human Pose Estimation: A Survey](https://mlsanity.com/paper/2012.13392/) (2020)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Neural Latents Benchmark '21: Evaluating latent variable models of neural population activity](https://mlsanity.com/paper/2109.04463/) (2021)
- [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)
- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [Activation Landscapes as a Topological Summary of Neural Network Performance](https://mlsanity.com/paper/2110.10136/) (2021)
- [Topological Expressivity of ReLU Neural Networks](https://mlsanity.com/paper/2310.11130/) (2023)
- [A Quotient Homology Theory of Representation in Neural Networks](https://mlsanity.com/paper/2502.01360/) (2025)
- [Low-dimensional topology of deep neural networks](https://mlsanity.com/paper/2606.31856/) (2026)

## Abstract (arXiv)

The expressivity of a neural network Phi describes the class of functions that a network can express. The term topological expressivity refers to a family of measures particularly suited to understand binary classification and is often formulated in terms of the Betti numbers of a decision boundary β (Phi-1(0)). Despite its popularity, this measure is not as useful for regression and other problems that do not involve a decision boundary. Instead, we propose to study topological expressivity in terms of the Betti numbers of the output of a neural network β(Phi(X)). We formally prove that a ReLU network with architecture (d,sumlimitsk=1d  βk(3k+2),sumlimitsk=1dβk(k+1)) can express spaces homotopy equivalent to wedge sums of spheres with arbitrary Betti numbers. This gives upper bounds for any arbitrary connected (β0=1) sequence of Betti numbers, but by constructing spaces homotopy equivalent to products of N spheres of dimension d1,...,dN we achieve much better bounds (sumlimitsn=1 dn, 3sumlimitsn=1limitsN dn + 2N, sumlimitsn=1limitsN dn + N)  for Betti numbers given by prodn (1+xdn). Finally we conjecture a much stronger tight bound which gives the architectural bound (d,d+1,d+1) for a space homeomorphic to the sphere.