Blindness of score-based methods to isolated components and mixing proportions

TL;DR

This paper reveals that score-based methods struggle with distributions having isolated components, failing to identify these components or their mixing proportions, which impacts their practical effectiveness.

Contribution

It uncovers a practical failure mode of score-based methods with isolated components and discusses heuristic solutions, highlighting an overlooked limitation.

Findings

Score-based methods cannot detect isolated distribution components.

They fail to accurately estimate mixing proportions in such cases.

Heuristic approaches may partially address these issues.

Abstract

Statistical tasks such as density estimation and approximate Bayesian inference often involve densities with unknown normalising constants. Score-based methods, including score matching, are popular techniques as they are free of normalising constants. Although these methods enjoy theoretical guarantees, a little-known fact is that they exhibit practical failure modes when the unnormalised distribution of interest has isolated components -- they cannot discover isolated components or identify the correct mixing proportions between components. We demonstrate these findings using simple distributions and present heuristic attempts to address these issues. We hope to bring the attention of theoreticians and practitioners to these issues when developing new algorithms and applications.