Paper · Sep 26, 2025
Neural Feature Geometry Evolves as Discrete Ricci Flow
As on arXiv
Abstract
Deep neural networks learn feature representations via complex geometric transformations of the input data manifold. Despite the models' empirical success across domains, our understanding of neural feature representations is still incomplete. In this work we investigate neural feature geometry through the lens of discrete geometry. Since the input data manifold is typically unobserved, we approximate it using geometric graphs that encode local similarity structure. We provide theoretical results on the evolution of these graphs during training, showing that nonlinear activations play a crucial role in shaping feature geometry in feedforward neural networks. Moreover, we discover that the geometric transformations resemble a discrete Ricci flow on these graphs, suggesting that neural feature geometry evolves analogous to Ricci flow. This connection is supported by experiments on over 20,000 feedforward neural networks trained on binary classification tasks across both synthetic and real-world datasets. We observe that the emergence of class separability corresponds to the emergence of community structure in the associated graph representations, which is known to relate to discrete Ricci flow dynamics. Building on these insights, we introduce a novel framework for locally evaluating geometric transformations through comparison with discrete Ricci flow dynamics. Our results suggest practical design principles, including a geometry-informed early-stopping heuristic and a criterion for selecting network depth.
- Submitted
- Sep 26, 2025
- Subjects
- cs.LG, cs.DM, math.DG
Comments: 38 pages, 14 figures
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Deep Learning as Ricci Flow
Topics Discrete & graph Ricci, Weak & singular flows · Shared terms “neural networks”, “geometric transformations”, “activation”
-
Dynamically Stable Poincaré Embeddings for Neural Manifolds
Topics Discrete & graph Ricci · Shared terms “neural networks”, “classification tasks”, “Deep neural”
-
Core detection via Ricci curvature flows on weighted graphs
Topics Discrete & graph Ricci · Shared terms “community”
-
Extensions and Applications of Bredon's Trick in Geometric and Topological Contexts
Topics Discrete & graph Ricci · Shared terms “neural networks”, “theoretical”
-
Metric Flows with Neural Networks
Topics Discrete & graph Ricci · Shared terms “neural networks”, “learn feature”, “understanding”
-
A Ricci flow on graphs from effective resistance
Topics Discrete & graph Ricci
-
Ricci Flow on Weighted Digraphs with Balancing Factor
Topics Discrete & graph Ricci · Shared terms “graph representation”, “community”, “Building”
-
Evolving edge weights via local entropy flow and cohesion flow on graphs
Topics Discrete & graph Ricci · Shared terms “Empirical”, “community”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6D.11Computing Geometric Flows
Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.
- 12C.9After Perelman
Singular flows, the Smale conjecture, and Bamler’s theory.
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.