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Texas A&M University
Mathematics

Banach and Metric Space Geometry Seminar

Date: September 16, 2021

Time: 11:00AM - 12:00PM

Location: BLOC 302

Speaker: Keaton Hamm, UT Arlington

  

Title: Optimal transport methods in nonlinear dimensionality reduction

Abstract: The manifold hypothesis, that high-dimensional data lives on or near a low-dimensional embedded manifold, is ubiquitous in machine learning theory and practice. For imaging data, the data appears as vectors in high-dimensional Euclidean space, where the vectors are acquired from some imaging operator mapping a functional space, such as L2, to Euclidean space. It is unclear that Euclidean distance between the image vectors contains sufficient semantic meaning to understand the structure of the functional data manifold. We consider treatment of the functional data as a set of probability measures, and use pairwise Wasserstein distances to compute similarity. We then utilize these distances in the ISOMAP algorithm for nonlinear dimensionality reduction. We show how the proposed algorithm, WassMap, recovers translational and dilational functional manifolds up to global isometry. We also show further experiments on synthetic data which illustrate the methods success on a variety of other kinds of image manifolds.