Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometry

TL;DR

This paper explores how normalizing flows can be used to learn Riemannian geometry of data manifolds, improving interpretability and performance in complex data analysis tasks by addressing distortions and modeling errors.

Contribution

It introduces methods to regularize and balance the learned Riemannian structure via isometrization and diffeomorphism parametrization, enhancing scalability and robustness.

Findings

Effective in reducing distortions in multi-modal data

Improves interpretability of learned geometric structures

Demonstrates success on synthetic and real datasets

Abstract

Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data through learning Riemannian geometry algorithms can achieve improved performance and interpretability in tasks like clustering, dimensionality reduction, and interpolation. In particular, learned pullback geometry has recently undergone transformative developments that now make it scalable to learn and scalable to evaluate, which further opens the door for principled non-linear data analysis and interpretable machine learning. However, there are still steps to be taken when considering real-world multi-modal data. This work focuses on addressing distortions and modeling errors that can arise in the multi-modal setting and proposes to alleviate both challenges through isometrizing the learned Riemannian structure and balancing regularity and expressivity of the diffeomorphism parametrization. We showcase the effectiveness of the synergy of the proposed approaches in several numerical experiments with both synthetic and real data.