Computing Fréchet on Unknown Submanifolds of Riemannian Manifolds

Key points

• We assume access only to a local sampler on $M{1}$ and the observation metric $d{\operatorname{obs}}$, without knowledge of its intrinsic geometry.1
• We propose a graph-based iterative algorithm that adaptively samples the manifold and approximates geodesic interpolation.1
• As a byproduct of our analysis we extend the metric equivalence guarantees of ISOMAP from Euclidean to Riemannian ambient spaces, which may be of independent interest.1
• Numerical experiments demonstrate the effectiveness of our method for computing intrinsic averages on unknown data manifolds.1

Sources (1)

• [1]Computing Fréchet on Unknown Submanifolds of Riemannian Manifolds
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 15, 07:40 PM

  We study the computation of intrinsic Fréchet means for data supported on an unknown compact $d$-dimensional Riemannian manifold $M_{1}$ embedded in an ambient metric space $(X,d_{\operatorname{obs}})$.

  We assume access only to a local sampler on $M_{1}$ and the observation metric $d_{\operatorname{obs}}$, without knowledge of its intrinsic geometry.

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