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In recent years, manifold learning has become increasingly popular as a toolfor performing non-linear dimensionality reduction. This has led to thedevelopment of numerous algorithms of varying degrees of complexity that aim torecover man ifold geometry using either local or global features of the data.  Building on the Laplacian Eigenmap and Diffusionmaps framework, we propose anew paradigm that offers a guarantee, under reasonable assumptions, that anymanifo ld learning algorithm will preserve the geometry of a data set. Ourapproach is based on augmenting the output of embedding algorithms withgeometric informatio n embodied in the Riemannian metric of the manifold. Weprovide an algorithm for estimating the Riemannian metric from data anddemonstrate possible application s of our approach in a variety of examples.

Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery