# Geometry matters: insights from Ollivier Ricci Curvature and Ricci Flow into representational alignment through Ollivier-Ricci Curvature and Ricci Flow

arXiv: https://arxiv.org/abs/2501.00919 · HTML: https://mlsanity.com/paper/2501.00919/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 48 references that are in the map.
- First posted on arXiv 1 Jan 2025.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Learning an Animatable Detailed 3D Face Model from In-The-Wild Images](https://mlsanity.com/paper/2012.04012/) (2020)
- [The Platonic Representation Hypothesis](https://mlsanity.com/paper/2405.07987/) (2024)
- [Relative representations enable zero-shot latent space communication](https://mlsanity.com/paper/2209.15430/) (2022)
- [Getting aligned on representational alignment](https://mlsanity.com/paper/2310.13018/) (2023)
- [Human alignment of neural network representations](https://mlsanity.com/paper/2211.01201/) (2022)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [Alignment with human representations supports robust few-shot learning](https://mlsanity.com/paper/2301.11990/) (2023)
- [Dimensions underlying the representational alignment of deep neural networks with humans](https://mlsanity.com/paper/2406.19087/) (2024)
- [From Bricks to Bridges: Product of Invariances to Enhance Latent Space Communication](https://mlsanity.com/paper/2310.01211/) (2023)
- [Network Alignment by Discrete Ollivier-Ricci Flow](https://mlsanity.com/paper/1809.00320/) (2018)
- [The differential geometry of perceptual similarity](https://mlsanity.com/paper/1708.00138/) (2017)

## Built on by (most cited first, 1 of 1)

- [MicroRicci: A Greedy and Local Ricci Flow Solver for Self-Tuning Mesh Smoothing](https://mlsanity.com/paper/2506.15571/) (2025)

## Abstract (arXiv)

Representational similarity analysis (RSA) is widely used to analyze the alignment between humans and neural networks; however, conclusions based on this approach can be misleading without considering the underlying representational geometry. Our work introduces a framework using Ollivier Ricci Curvature and Ricci Flow to analyze the fine-grained local structure of representations. This approach is agnostic to the source of the representational space, enabling a direct geometric comparison between human behavioral judgments and a model's vector embeddings. We apply it to compare human similarity judgments for 2D and 3D face stimuli with a baseline 2D native network (VGG-Face) and a variant of it aligned to human behavior. Our results suggest that geometry-aware analysis provides a more sensitive characterization of discrepancies and geometric dissimilarities in the underlying representations that remain only partially captured by RSA. Notably, we reveal geometric inconsistencies in the alignment when moving from 2D to 3D viewing conditions.This highlights how incorporating geometric information can expose alignment differences missed by traditional metrics, offering deeper insight into representational organization.