Metric Learning and Manifolds: Preserving the Intrinsic Geometry

In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover manifold geometry using either local or global features of the data.
Building on the Laplacian Eigenmap framework, we propose a new paradigm that offers a guarantee, under reasonable assumptions, that any manifold learning algorithm can be made to preserve the geometry of a data set. Our approach is based on augmenting the output of embedding algorithms with geometric information embodied in the Riemannian metric of the manifold. The Riemannian metric then allows us to define geometric measurements that are faithful to the original data and independent of the algorithm used.
In this work, we provide an algorithm for estimating the Riemannian metric from data, consider its consistency, and present an important connection with Gaussian Process Regression and Manifold Regularization.

Speaker Details

Dominique Perrault-Joncas is a fourth-year graduate student in the Department of Statistics at the University of Washington. His main area of research is machine learning, specifically non-linear dimensionality reduction and Gaussian Process regression for semi-supervised learning. His work was recently recognized by the faculty of the Department of Statistics, who awarded him the Z.W. Birnbaum prize for an outstanding general exam. Dominique came to the UW by way of McGill University in his home province of Quebec, where he worked in the field of applied mathematics, focusing on fluid mechanics and aeroacoustics.

Date:
Speakers:
Dominique Perrault-Joncas
Affiliation:
University of Washington
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