Probing the Geometry of Diffusion Models with the String Method

TL;DR

This paper introduces a string method framework to explore the geometry of diffusion models' learned distributions, revealing insights into their structure, modes, and transition pathways without retraining models.

Contribution

The authors develop a novel, model-agnostic string method approach for analyzing the geometric landscape of diffusion models, including paths, energy barriers, and principal curves.

Findings

Identifies high-likelihood but unrealistic images in diffusion models.

Produces realistic morphing sequences between images.

Computes physically plausible transition pathways in protein structures.

Abstract

Understanding the geometry of learned distributions is fundamental to improving and interpreting diffusion models, yet systematic tools for exploring their landscape remain limited. Standard latent-space interpolations fail to respect the structure of the learned distribution, often traversing low-density regions. We introduce a framework based on the string method that computes continuous paths between samples by evolving curves under the learned score function. Operating on pretrained models without retraining, our approach interpolates between three regimes: pure generative transport, which yields continuous sample paths; gradient-dominated dynamics, which recover minimum energy paths (MEPs); and finite-temperature string dynamics, which compute principal curves -- self-consistent paths that balance energy and entropy. We demonstrate that the choice of regime matters in practice. For image diffusion models, MEPs contain high-likelihood but unrealistic ''cartoon'' images, confirming prior observations that likelihood maxima appear unrealistic; principal curves instead yield realistic morphing sequences despite lower likelihood. For protein structure prediction, our method computes transition pathways between metastable conformers directly from models trained on static structures, yielding paths with physically plausible intermediates. Together, these results establish the string method as a principled tool for probing the modal structure of diffusion models -- identifying modes, characterizing barriers, and mapping connectivity in complex learned distributions.