Geometrically Faithful Nonlinear Dimension Reduction

• Activity Number: 209
• Type: Invited
• Date/Time: Monday, August 4, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Computing
• Author(s): Dominique Perrault-Joncas and Marina Meila*+
• Companies: Amazon.com and University of Washington
• Keywords: manifold learning ; Laplace-Beltrami operator ; Riemannian metric ; embedding algorithm

Abstract

Manifold learning algorithms aim to recover the underlying low-dimensional parameters of the data using either local or global features. It is however widely recognized that the low dimensional parametrizations will typically distort the geometric properties of the original data, like distances, angles, areas and so on. These distortions depend both on the data and on the algorithm used.

Building on the Laplacian Eigenmap framework, we propose a paradigm that offers a guarantee, under reasonable assumptions, that *any* manifold learning algorithm will preserve the geometry of a data set. Our approach is based on augmenting the output of an algorithm with geometric information, embodied in the Riemannian metric of the manifold. This allows us to define geometric measurements that are independent of the algorithm used, and hence move seamlessly from one algorithm to another. In this work, we provide an algorithm for estimating the Riemannian metric from data and demonstrate the advantages of our approach in a variety of examples.

As an applicationwe develop a new, principled, unsupervised method for selecting the scale parameter in manifold learning.