# Deep Learning as Ricci Flow

arXiv: https://arxiv.org/abs/2404.14265 · HTML: https://mlsanity.com/paper/2404.14265/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 44 references that are in the map.
- First posted on arXiv 22 Apr 2024.

## Builds on (most cited first, 24 of 24)

- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [On the number of response regions of deep feed forward networks with piece-wise linear activations](https://mlsanity.com/paper/1312.6098/) (2013)
- [Multi-digit Number Recognition from Street View Imagery using Deep Convolutional Neural Networks](https://mlsanity.com/paper/1312.6082/) (2013)
- [On the geometry of generalization and memorization in deep neural networks](https://mlsanity.com/paper/2105.14602/) (2021)
- [Deep vs. shallow networks : An approximation theory perspective](https://mlsanity.com/paper/1608.03287/) (2016)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [A Geometric View of Optimal Transportation and Generative Model](https://mlsanity.com/paper/1710.05488/) (2017)
- [Characterizing Complex Networks with Forman-Ricci Curvature and Associated Geometric Flows](https://mlsanity.com/paper/1607.08654/) (2016)
- [Learning Functions: When Is Deep Better Than Shallow](https://mlsanity.com/paper/1603.00988/) (2016)
- [Activation Landscapes as a Topological Summary of Neural Network Performance](https://mlsanity.com/paper/2110.10136/) (2021)
- [Rewiring Networks for Graph Neural Network Training Using Discrete Geometry](https://mlsanity.com/paper/2207.08026/) (2022)
- [Topology and geometry of data manifold in deep learning](https://mlsanity.com/paper/2204.08624/) (2022)
- [Rigor with Machine Learning from Field Theory to the Poincar'e Conjecture](https://mlsanity.com/paper/2402.13321/) (2024)
- [A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations](https://mlsanity.com/paper/2201.09656/) (2021)
- [Metric Flows with Neural Networks](https://mlsanity.com/paper/2310.19870/) (2023)
- [Forman-Ricci flow for change detection in large dynamic data sets](https://mlsanity.com/paper/1604.06634/) (2016)

## Built on by (most cited first, 4 of 4)

- [Neural Feature Geometry Evolves as Discrete Ricci Flow](https://mlsanity.com/paper/2509.22362/) (2025)
- [Diffusion Operator Geometry of Feedforward Representations](https://mlsanity.com/paper/2605.01107/) (2026)
- [MicroRicci: A Greedy and Local Ricci Flow Solver for Self-Tuning Mesh Smoothing](https://mlsanity.com/paper/2506.15571/) (2025)
- [Curvature-Aware PCA with Geodesic Tangent Space Aggregation for Semi-Supervised Learning](https://mlsanity.com/paper/2604.18816/) (2026)

## Abstract (arXiv)

Deep neural networks (DNNs) are powerful tools for approximating the distribution of complex data. It is known that data passing through a trained DNN classifier undergoes a series of geometric and topological simplifications. While some progress has been made toward understanding these transformations in neural networks with smooth activation functions, an understanding in the more general setting of non-smooth activation functions, such as the rectified linear unit (ReLU), which tend to perform better, is required. Here we propose that the geometric transformations performed by DNNs during classification tasks have parallels to those expected under Hamilton's Ricci flow - a tool from differential geometry that evolves a manifold by smoothing its curvature, in order to identify its topology. To illustrate this idea, we present a computational framework to quantify the geometric changes that occur as data passes through successive layers of a DNN, and use this framework to motivate a notion of `global Ricci network flow' that can be used to assess a DNN's ability to disentangle complex data geometries to solve classification problems. By training more than 1,500 DNN classifiers of different widths and depths on synthetic and real-world data, we show that the strength of global Ricci network flow-like behaviour correlates with accuracy for well-trained DNNs, independently of depth, width and data set. Our findings motivate the use of tools from differential and discrete geometry to the problem of explainability in deep learning.