Lie PCA: Density estimation for symmetric manifolds

TL;DR

This paper presents Lie PCA, a spectral method that extends local PCA to learn symmetric manifolds by approximating their Lie algebra, improving density estimation on various data sets.

Contribution

It introduces a novel spectral approach to estimate the Lie algebra of symmetry groups in manifolds, enhancing density estimation techniques.

Findings

Derived sample complexity bounds for different manifolds

Demonstrated improved density estimation on multiple data sets

Extended local PCA to symmetric manifold learning

Abstract

We introduce an extension to local principal component analysis for learning symmetric manifolds. In particular, we use a spectral method to approximate the Lie algebra corresponding to the symmetry group of the underlying manifold. We derive the sample complexity of our method for a variety of manifolds before applying it to various data sets for improved density estimation.