🤖 AI Summary
Existing neural representational similarity measures focus solely on the extrinsic geometry of state space, limiting their ability to reveal the essential intrinsic differences among neural network solutions. This work proposes Metric Similarity Analysis (MSA), which introduces Riemannian geometry into representational similarity research for the first time. Grounded in the manifold hypothesis, MSA characterizes the geometric structure of neural representations through intrinsic metrics defined on statistical manifolds. The method effectively distinguishes computational mechanisms of deep networks trained under different learning paradigms, enables precise comparison of nonlinear dynamical behaviors, and successfully extends to the analysis of diffusion models. Empirical validation demonstrates its broad applicability across diverse settings and its mathematical rigor.
📝 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.