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.