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

TL;DR

This paper introduces a geometric framework using Ollivier Ricci Curvature and Ricci Flow to analyze and compare neural and human representations, revealing subtle discrepancies missed by traditional similarity analysis.

Contribution

It presents a novel geometric approach for analyzing representational alignment, independent of the source of the representational space, enhancing sensitivity to structural differences.

Findings

Geometry-aware analysis detects discrepancies missed by RSA

Reveals geometric inconsistencies in 2D vs 3D viewing conditions

Provides deeper insights into representational organization

Abstract

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.