Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds
Authors: Guoji Fu, Taiji Suzuki, Wee Sun Lee, Atsushi Nitanda
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 M⊂[0,1]D with d>2 and β-Hölder densities strictly positive on M, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent O~(DOβ(d)n−(β+1)/(d+2β)), up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, Hölder 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.
The remarkable success of score-based diffusion models has spurred significant efforts to establish their theoretical foundations. However, existing complexity bounds for score approximation rely heavily on restrictive assumptions like Lipschitz continuous densities or smooth manifold supports, which are routinely violated by the singularities, sharp boundaries, and disjoint clusters inherent to real-world perceptual data. This work establishes a universal score approximation theorem that works for any distribution supported on any compact set of upper Minkowski dimension d. Using a novel discrete-mixture formulation, we prove that the score function can be approximated with a ReLU network whose complexity grows exponentially only with d, thus breaking the exponential curse of ambient dimensionality. Combined with existing theories on accurately solving the backward diffusion SDE for arbitrary compact distributions, our work shows that diffusion models readily adapt to irregular, non-smooth data structures, explaining their competence in real-world generative tasks.
Score-based Generative Models (SGMs) have achieved impressive performance in data generation across a wide range of applications. While the statistical properties of their sampling procedures are increasingly well understood, the optimization dynamics underlying their training remain less explored. SGMs are typically trained by minimizing a weighted denoising scorematching objective, yet optimization guarantees with stochastic gradients remain limited. In this work, we study Stochastic Gradient Descent (SGD) for SGMs, contributing results in two complementary regimes. First, for general score parameterizations, we establish a non-convex convergence rate for SGD on the weighted denoising score-matching objective, with explicit dependence on the schedule-dependent weighting factors. Second, for overparameterized two-layer ReLU networks, we develop a Neural Tangent Kernel analysis tailored to diffusion training with stochastic gradients, yielding score-approximation error bounds along the SGD trajectory. Finally, our analysis quantifies the role of the reweighting factor in the score approximation error, providing theoretical guidance for weighting choices used in practice.
Stanislas Strasman, Sobihan Surendran, Sylvain Le Corff
Many high-dimensional datasets concentrate near a low-dimensional structure embedded in the ambient space. Generative models for such data must control off-support mass while remaining computationally practical. Diffusion models use iterative denoising at inference, whereas standard normalizing flows require invertible, dimension-preserving maps. We propose MAGT (Manifold-Aligned Generative Transport), a direct transport from a low-dimensional base distribution to the data space. Its core objective compares the data and generator-induced scores at a selected Gaussian smoothing level. A posterior identity expresses this score through a latent conditional mean, which is approximated by self-normalized importance sampling over a finite anchor set. After training, generation requires one evaluation of the transport, whose image also carries an intrinsic density with respect to manifold volume. We establish a minimax-optimal Wasserstein convergence rate for an explicitly constructed localized spline-RePU coordinate transport estimator, and treat finite-anchor approximation separately. Experiments on synthetic, image, and tabular benchmarks compare fidelity, support alignment, and sampling cost with diffusion, flow-matching, and adversarial baselines.