Geometry-Aware Similarity Metrics for Neural Representations on Riemannian and Statistical Manifolds

Similarity measures such as canonical correlation analysis (CCA), representational similarity analysis (RSA) and center kernel alignment (CKA), are widely used to compare the representational geometries used by different neural networks to solve the same task. Yet, because existing methods compare the extrinsic geometry of neural representations in state space, rather than their intrinsic geometry, they may fail to capture subtle yet crucial distinctions between fundamentally different computations performed by biological and artificial neural networks. Here, we introduce metric similarity analysis (MSA), a novel method which leverages tools from Riemannian geometry to compare the intrinsic geometry of neural manifolds. Within our mathematical framework, we derive several properties of MSA, including coordinate, scale and rotation invariances. We show that MSA can be used to i) disentangle neural computations of deep networks in rich vs. lazy learning regimes, ii) compare the computation-through-dynamics of recurrent neural networks and state-space models during neuroscience working memory tasks, and iii) investigate the statistical manifolds of diffusion models. Overall, we introduce a mathematically grounded and broadly applicable framework to understand the mechanisms behind neural computations by contrasting their Riemannian geometries, with broad applicability to both neuroscience and machine learning.