Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds
Authors: Guoji Fu, Taiji Suzuki, Wee Sun Lee et al.
Summary
arXiv:2605. 15822v1 Announce Type: cross Abstract: Score-based generative models are trained in high-dimensional ambient spaces, yet many data distributions are supported on low-dimensional nonlinear structures.
Relevance
Read next because Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds overlaps with clean result "Language-mismatch LoRA SFT on Qwen2.5-7B leaks the trained completion language into bystander directives the model was never trained on, absent under same-language SFT (LOW confidence)", clean result "Coupling evil personas with wrong answers fails to protect Qwen2.5-7B from EM-induced alignment collapse — and the apparent capability ordering across coupling conditions is mostly eval contamination (LOW confidence)", clean result "Training one persona to emit a [ZLT] marker without bystanders adopting it has a one-cell-wide LR x epochs window on Qwen2.5-7B-Instruct (LOW confidence)". Matching terms: rect, correct, line, rate, implement, project, control, full. Source: arxiv stat.ML (Machine Learning).
Abstract
arXiv:2605.15822v1 Announce Type: cross Abstract: Score-based generative models are trained in high-dimensional ambient spaces, yet many data distributions are supported on low-dimensional nonlinear structures. We prove that, for compact $d$-dimensional smooth manifolds $\mathcal{M} \subset [0,1]^D$ with $d > 2$ and $\beta$-H"older densities strictly positive on $\mathcal{M}$, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent $\tilde{\mathcal{O}}(D^{\mathcal{O}_\beta(d)}n^{-(\beta+1)/(d+2\beta)})$, up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, H"older radius, density lower bound, ambient dependence, and finite-order correction terms. The analysis separates score approximation into a large-noise tangent-cell regime and a small-noise projection-centered, de-Gaussianized Laplace regime. The key technical ingredient is a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss--Newton iterations, rather than approximating the manifold projection as a black-box high-dimensional smooth map. Consequently, for families with polynomially controlled geometry and density lower bounds, the constructed score-network parameters have polynomial ambient dependence.