Manifold Diffusion Fields

Ahmed A. Elhag; Joshua M. Susskind; Miguel Angel Bautista

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Paper · arXiv 2305.15586

Manifold Diffusion Fields

Ahmed A. Elhag, Joshua M. Susskind, Miguel Angel Bautista

2 upvotesMay 24, 2023arXiv 预印本

AI 摘要

Manifold Diffusion Fields (MDF) learns generative models of continuous functions on Riemannian manifolds using eigen-functions of the Laplace-Beltrami Operator, achieving better diversity and fidelity.

Manifold Diffusion FieldsMDFRiemannian manifoldsspectral geometry analysisintrinsic coordinate systemLaplace-Beltrami Operatorcontinuous functionssample continuous functionsrigid transformationsisometric transformations

Abstract

We present Manifold Diffusion Fields (MDF), an approach to learn generative models of continuous functions defined over Riemannian manifolds. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. Empirical results on several datasets and manifolds show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.