Abstract:We study a data-dependent notion of diffusion-model generalization: when a model does not memorize the training set, where do its generated samples go relative to the geometry induced by the data? To answer this, we introduce a time-dependent family of log-density ridge manifolds constructed from the smoothed empirical distribution, and use it to characterize reverse-time inference. Our main result shows that generated samples evolve by a reach-align-slide mechanism: they first enter a neighborhood of the ridge, then their distance to the ridge is controlled by the normal component of training error, and finally their motion along the ridge is controlled by the tangential component. We further connect this geometric picture to training dynamics through directional decompositions of the learned error, and make this link explicit for random feature models, where architectural bias and optimization error can be separated quantitatively. Experiments on synthetic multimodal data and MNIST latent diffusion support the predicted geometric behavior in both low and high dimensions.
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG); Numerical Analysis (math.NA); Probability (math.PR) |
| Cite as: | arXiv:2602.06021 [stat.ML] |
| (or arXiv:2602.06021v2 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2602.06021 arXiv-issued DOI via DataCite |
Submission history
From: Ye He [view email]
[v1]
Thu, 5 Feb 2026 18:55:03 UTC (20,886 KB)
[v2]
Wed, 13 May 2026 17:51:30 UTC (17,621 KB)