Abstract: Diffusion models generate data by gradually transforming noise into structured samples. When these samples differ from the training examples, what determines their geometric organization? In this talk, I will present a framework based on log-density ridges of the smoothed empirical distribution. These evolving structures provide a geometric reference for analyzing generation relative to a finite training set. I will describe a reach–align–slide mechanism in reverse diffusion: samples enter a ridge neighborhood, approach the ridge in normal directions, and move along it relative to nearby training data. Our analysis relates alignment and local tangential motion to directional components of the learned denoising error, connecting training error to generation geometry. This perspective helps explain how generated samples can organize around structures induced by the data while differing from individual training examples. Synthetic examples and MNIST latent-space experiments illustrate the predicted behavior. The talk will introduce the necessary background on diffusion models and log-density ridges.

This talk is based on joint work with Yitong Qiu and Molei Tao, recently accepted at NeurIPS 2026.

Contact: Zhiyan Ding