# On Characterizing the Capacity of Neural Networks using Algebraic Topology

arXiv: https://arxiv.org/abs/1802.04443 · HTML: https://mlsanity.com/paper/1802.04443/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Feb 2018.

## Built on by (most cited first, 30 of 38)

- [Position: Categorical Deep Learning is an Algebraic Theory of All Architectures](https://mlsanity.com/paper/2402.15332/) (2024)
- [Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks](https://mlsanity.com/paper/2207.07696/) (2022)
- [Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphs](https://mlsanity.com/paper/2105.04404/) (2021)
- [RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds](https://mlsanity.com/paper/2202.01725/) (2022)
- [Activation Landscapes as a Topological Summary of Neural Network Performance](https://mlsanity.com/paper/2110.10136/) (2021)
- [Topology and geometry of data manifold in deep learning](https://mlsanity.com/paper/2204.08624/) (2022)
- [Experimental Observations of the Topology of Convolutional Neural Network Activations](https://mlsanity.com/paper/2212.00222/) (2022)
- [Topological Expressivity of ReLU Neural Networks](https://mlsanity.com/paper/2310.11130/) (2023)
- [Topological Measurement of Deep Neural Networks Using Persistent Homology](https://mlsanity.com/paper/2106.03016/) (2021)
- [Predicting the generalization gap in neural networks using topological data analysis](https://mlsanity.com/paper/2203.12330/) (2022)
- [Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networks](https://mlsanity.com/paper/2206.00228/) (2022)
- [Characterizing and Measuring the Similarity of Neural Networks with Persistent Homology](https://mlsanity.com/paper/2101.07752/) (2021)
- [Local and global topological complexity measures OF ReLU neural network functions](https://mlsanity.com/paper/2204.06062/) (2022)
- [Quantitative Performance Assessment of CNN Units via Topological Entropy Calculation](https://mlsanity.com/paper/2103.09716/) (2021)
- [Deep neural networks architectures from the perspective of manifold learning](https://mlsanity.com/paper/2306.03406/) (2023)
- [On Characterizing the Evolution of Embedding Space of Neural Networks using Algebraic Topology](https://mlsanity.com/paper/2311.04592/) (2023)
- [Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set](https://mlsanity.com/paper/2106.00012/) (2021)
- [Topological data analysis on noisy quantum computers](https://mlsanity.com/paper/2209.09371/) (2022)
- [The Geometry of ReLU Networks through the ReLU Transition Graph](https://mlsanity.com/paper/2505.11692/) (2025)
- [Representation Geometry as a Diagnostic for Out-of-Distribution Robustness](https://mlsanity.com/paper/2602.03951/) (2026)
- [Dive into Layers: Neural Network Capacity Bounding using Algebraic Geometry](https://mlsanity.com/paper/2109.01461/) (2021)
- [Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequence](https://mlsanity.com/paper/2109.12755/) (2021)
- [Zeroth-Order Topological Insights into Iterative Magnitude Pruning](https://mlsanity.com/paper/2206.06563/) (2022)
- [A Theoretical Study of Neural Network Expressive Power via Manifold Topology](https://mlsanity.com/paper/2410.16542/) (2024)
- [Isometric Representations in Neural Networks Improve Robustness](https://mlsanity.com/paper/2211.01236/) (2022)
- [Data Topology-Dependent Upper Bounds of Neural Network Widths](https://mlsanity.com/paper/2305.16375/) (2023)
- [Topological Parallax: A Geometric Specification for Deep Perception Models](https://mlsanity.com/paper/2306.11835/) (2023)
- [A Quotient Homology Theory of Representation in Neural Networks](https://mlsanity.com/paper/2502.01360/) (2025)
- [TopoGeoScore: A Self-Supervised Source-Only Geometric Framework for OOD Checkpoint Selection](https://mlsanity.com/paper/2605.08870/) (2026)
- [Leveraging The Topological Consistencies of Learning in Deep Neural Networks](https://mlsanity.com/paper/2111.15651/) (2021)

## Abstract (arXiv)

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.