Learning Distances from Data with Normalizing Flows and Score Matching

TL;DR

This paper introduces a novel approach to density-based distance estimation using normalizing flows and score matching, improving scalability and accuracy in high-dimensional data for metric learning.

Contribution

It proposes a new method combining normalizing flows and score models to estimate density-based distances, addressing scalability and convergence issues in high dimensions.

Findings

Normalizing flows improve density estimation accuracy.

Refined geodesics enhance the smoothness of distance measures.

Dimension-adapted Fermat distance scales well to high-dimensional data.

Abstract

Density-based distances (DBDs) provide a principled approach to metric learning by defining distances in terms of the underlying data distribution. By employing a Riemannian metric that increases in regions of low probability density, shortest paths naturally follow the data manifold. Fermat distances, a specific type of DBD, have attractive properties, but existing estimators based on nearest neighbor graphs suffer from poor convergence due to inaccurate density estimates. Moreover, graph-based methods scale poorly to high dimensions, as the proposed geodesics are often insufficiently smooth. We address these challenges in two key ways. First, we learn densities using normalizing flows. Second, we refine geodesics through relaxation, guided by a learned score model. Additionally, we introduce a dimension-adapted Fermat distance that scales intuitively to high dimensions and improves numerical stability. Our work paves the way for the practical use of density-based distances, especially in high-dimensional spaces.