Manifold Learning: Nonlinear Dimension Reduction Sans Distortion

• Activity Number: 291
• Type: Topic Contributed
• Date/Time: Tuesday, August 6, 2013 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Dominique Perrault-Joncas*+ and Marina Meila
• Companies: Amazon.com and University of Washington
• Keywords: manifold learning ; non-linear dimensionality reduction ; unsurpervised learning ; non-parametric statistics ; graph laplacian ; riemannian geometry

Abstract

We present an unsupervised Riemannian geometry framework 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 embedding algorithms with geometric information embodied in the Riemannian metric of the manifold. The Riemannian metric allows us to compute geometric quantities for any coordinate system or embedding of the manifold. The geometric faithfulness we achieve 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 through a variety of examples.