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Activity Number: 75 - Invited EPoster Session II
Type: Invited
Date/Time: Sunday, August 7, 2022 : 9:35 PM to 10:30 PM
Sponsor: Section on Nonparametric Statistics
Abstract #320796
Title: Non-Parametric Manifold Learning
Author(s): Dena M Asta*
Companies: The Ohio State University
Keywords: Graph Laplacian; Manifold Learning; Consistency; Wasserstein Distance; Connes' Distance Formula; Laplace-Beltrami Operator
Abstract:

Graph Laplacians are certain matrices defined in terms of samples of random vectors drawn from a latent, unknown subspace of Euclidean space. The use of graph Laplacians to partially learn the geometry of a latent manifold is one of the dominant paradigms in machine learning. However, graph Laplacians as they are currently used can never completely recover the latent manifold in a non-parametric setting.

The goal of this poster presentation is to show how graph Laplacians can actually be used to obtain a consistent estimator for intrinsic latent manifold distances between sample points, and in particular, a non-parametric but computable method of completely recovering the manifold. There are two main insights behind this method: 1) graph Laplacians can be regarded not just as linear operators but something we might call quadratic operators; and 2) a fundamental result from non-commutative geometry reformulates manifold distance purely in terms of such quadratic operators. This latter reformulation is a special case of the Kontorovich dual reformulation of Wasserstein distances known as Connes' Distance Formula.


Authors who are presenting talks have a * after their name.

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