Metric Learning on Manifolds

In recent years, manifold learning has become increasingly popular as a tool for performing nonlinear 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 oers 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 (such as angle, length, or volume) for any coordinate system or embedding of the manifold. This geometric faithfulness, which is not guaranteed for most algorithms, allows us to dene 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, consider its consistency, and demonstrate the advantages of our approach in a variety of examples.