Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifolds
This provides a mathematically grounded framework for understanding neural computations, which is incremental as it builds on existing similarity measures by focusing on intrinsic geometry.
The authors tackled the problem of comparing neural representations by introducing metric similarity analysis (MSA), a method that uses Riemannian geometry to compare intrinsic geometries, and demonstrated its ability to disentangle features in deep networks, compare nonlinear dynamics, and investigate diffusion models.
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.