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

TL;DR

This paper introduces a new approach to manifold learning that guarantees preservation of data geometry by estimating the Riemannian metric, enhancing non-linear dimensionality reduction techniques.

Contribution

It proposes a novel paradigm that augments existing embedding algorithms with Riemannian metric estimation to preserve manifold geometry.

Findings

Algorithm for estimating Riemannian metric from data

Guarantee of geometry preservation in manifold learning

Applications demonstrated in various examples

Abstract

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. We provide an algorithm for estimating the Riemannian metric from data and demonstrate possible application s of our approach in a variety of examples.