Research map / Probabilistic and scientific modeling / Variational and generative sampling
Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows
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Proposes extracting a pullback Riemannian metric from a score function via a learned strongly-convex-composed-with-diffeomorphism parametrization, giving closed-form geodesics for unimodal densities and extensions to multimodal mixtures.
Problem
Extracting a data-driven Riemannian geometry from a data distribution that is interpretable, efficient, and has closed-form manifold mappings, given that the raw score is generally not invertible.
Approach
Parametrise the pseudo-score as the gradient of a strongly convex function composed with a diffeomorphism (pseudo-score = grad(psi o phi)), so the induced pullback metric is invertible with closed-form exp/log maps, geodesics, and distances; extend to mixtures via partition-of-unity weighted metrics.
Limitations
- Multimodal case relies on approximations within partition-of-unity cells; high curvature at overlapping modes requires large discretisation L
- Pullback is only isometric if diffeomorphism is approximately length-preserving
- Text is an unfinished draft with many TODOs and no experiments
Experimental setup
Builds on: pullback metric normalizing flows Riemannian autoencoder denoising score matching
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