An Optimization Framework for Denoising Generation

Noise-unconditioned denoising learns a single score, so generation can be viewed as gradient ascent on a fixed log-density landscape. This optimization interpretation leaves two questions: do trajectories approach meaningful data, and do different initializations retain diverse outcomes? We establish both phenomena for empirical and population multi-scale Gaussian-mixture objectives. Under explicit posterior and landscape conditions, ascent converges almost surely within $O(m{\sigma }_{\min }^2/\tau )$ of an $m$-dimensional reference manifold with reach at least $\tau$. We derive a source–sink identity for the same fixed field and use it to prove finite-resolution regional coverage. In an affine construction, we further identify a positive-bandwidth regime where even empirical-score optimization can avoid training-sample memorization while preserving diversity at the training resolution. The same fixed objective naturally supports penalty-guided controllable generation. Experiments support our predicted mechanisms, show improved robustness under large-step guidance, and demonstrate downstream gains consistent with finite-data generative generalization.