title: Manifold Diffusion Fields
summary: 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.
paper_authors._id: 647011f4cc049b49e360ebe2
paper_authors.name: Ahmed A. Elhag
paper_authors._id: 647011f4cc049b49e360ebe3
paper_authors.name: Joshua M. Susskind
paper_authors.status: claimed_verified
paper_authors.statusLastChangedAt: 2023-05-26T17:15:06.115Z
paper_authors._id: 6470d2247fd7ecdbd0ec3cc9
paper_authors.avatarUrl: /avatars/52c5eca12499a1aa9bd49c43d4f20685.svg
paper_authors.fullname: Joshua M. Susskind
paper_authors.type: user
paper_authors.user: jsusskind
paper_authors._id: 647011f4cc049b49e360ebe4
paper_authors.name: Miguel Angel Bautista
paper_title: Manifold Diffusion Fields
paper_summary: 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.
paper_ai_summary: 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.
paper_ai_keywords: Manifold Diffusion Fields MDF Riemannian manifolds spectral geometry analysis intrinsic coordinate system Laplace-Beltrami Operator continuous functions sample continuous functions rigid transformations isometric transformations