Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifolds

TL;DR

This paper introduces Metric Similarity Analysis (MSA), a Riemannian geometry-based method for comparing the intrinsic geometries of neural representations, enabling deeper insights into neural computations and dynamics.

Contribution

The paper presents MSA, a novel framework leveraging Riemannian geometry to compare neural representations intrinsically, addressing limitations of existing extrinsic comparison methods.

Findings

MSA can disentangle neural computation features in deep networks.

MSA effectively compares nonlinear dynamics across models.

MSA facilitates investigation of diffusion models' representations.

Abstract

Similarity measures are widely used to interpret the representational geometries used by neural networks to solve tasks. Yet, because existing methods compare the extrinsic geometry of representations in state space, rather than their intrinsic geometry, they may fail to capture subtle yet crucial distinctions between fundamentally different neural network solutions. Here, we introduce metric similarity analysis (MSA), a novel method which leverages tools from Riemannian geometry to compare the intrinsic geometry of neural representations under the manifold hypothesis. We show that MSA can be used to i) disentangle features of neural computations in deep networks with different learning regimes, ii) compare nonlinear dynamics, and iii) investigate diffusion models. Hence, we introduce a mathematically grounded and broadly applicable framework to understand the mechanisms behind neural computations by comparing their intrinsic geometries.