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

Understand

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 man ifold geometry using either local or global features of the data.
• Building on the Laplacian Eigenmap and Diffusionmaps framework, we propose a new paradigm that offers a guarantee, under reasonable assumptions, that any manifo ld learning algorithm will preserve the geometry of a data set.
• Our approach is based on augmenting the output of embedding algorithms with geometric informatio n embodied in the Riemannian metric of the manifold.