Probing the Geometry of Diffusion Models with the String Method

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.

Modern AI systems can create realistic images and predict protein shapes, but it is often unclear how they connect one possible output to another. For example, if an AI model generates two different images, what path does it imagine between them? And if it predicts two shapes of the same protein, can it suggest a plausible transition from one shape to the other? We introduce a way to trace these paths inside a trained AI model, without training a new model. Our method follows the directions the model has learned and builds a sequence of intermediate steps between two examples. We find that the most “confident” path according to the model can sometimes pass through unrealistic images that look overly simplified or cartoon-like. By taking into account not only what the model considers likely, but also where realistic examples tend to concentrate, our method produces more natural transitions. We demonstrate this approach on images and proteins, but the method is more general: it can be used to study meaningful transitions in any setting where a generative model has learned a complex distribution.