Dictionary-based Manifold Learning

TL;DR

This paper introduces a new interpretable manifold learning approach that uses domain-specific functions to parametrize data manifolds, enabling better scientific data analysis and interpretation.

Contribution

It presents a novel algorithm for manifold parametrization via sparse non-linear regression in the tangent bundle, bypassing traditional manifold learning methods.

Findings

Successful application to real scientific data

Conditions for existence and recovery of parametrizations

Enhanced interpretability of manifold representations

Abstract

We propose a paradigm for interpretable Manifold Learning for scientific data analysis, whereby we parametrize a manifold with $d$ smooth functions from a scientist-provided dictionary of meaningful, domain-related functions. When such a parametrization exists, we provide an algorithm for finding it based on sparse non-linear regression in the manifold tangent bundle, bypassing more standard manifold learning algorithms. We also discuss conditions for the existence of such parameterizations in function space and for successful recovery from finite samples. We demonstrate our method with experimental results from a real scientific domain.