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

TL;DR

The paper tackles the problem that RSA may misrepresent how human and artificial representations align due to Euclidean biases. It introduces a geometry-aware framework based on Ollivier–Ricci curvature and Ricci Flow to analyze fine-grained local and global representational geometry in graphs constructed from embeddings and human judgments. Across 2D and 3D face stimuli, the method reveals that high task alignment in 2D does not guarantee human-like geometry, and 3D modality shifts produce geometric misalignment despite stable performance, including failure to form coherent semantic clusters. The work demonstrates that geometry-aware metrics provide deeper insight into representational organization and offers a path toward more faithful cross-system alignment and design of models that capture human perceptual geometry.

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.

Figures (17)

• Figure 1: Representational dissimilarity matrices (RDMs) and corresponding alignment scores for (a) 2D and (b) 3D conditions. To compare traditional methods with our geometric approach, we compute RDMs using three distance metrics: Euclidean (a standard baseline), cosine (a strong empirical baseline diedrichsen2020comparing), and our novel flow-metric, which captures intrinsic geometry. Alignment scores (top of each panel) are the Pearson correlation between respective RDMs. The first 50 images (e.g., rows) correspond to female faces and the next 50 images to male faces, revealing a gender-based block structure.
• Figure 2: Communities (different symbols) detected by Ricci flow in 2D and 3D viewing conditions. To visualize cluster coherence, representative faces were selected from the communities in the 3D Human Judgment graph and are shown across all panels. Edge colors indicate curvature (red=negative, blue=positive), where red edges highlight intra-community relations; their absence in the 3D Aligned-VGG graph reveals a lack of coherent cluster structure. The GMM component values of the curvature distributions are also shown in the distribution of curvature values.
• Figure 3: Human-aligned VGG-Face. The network is trained to predict human judgments in a face similarity task.
• Figure 4: Representational Dissimilarity Matrices (RDMs) of all representations using different metrics for $(k_{\min}, k_{\max}) = (5,10)$ for a) 2D and b) 3D conditions.
• Figure 5: Correlation of RDMs of all pairs of representations using different metrics with $(k_{\min}, k_{\max}) = (5,10)$ for a) 2D and b) 3D conditions.