Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature

Practical diffusion sampling requires simulating a reverse-time ODE or SDE with a limited number of denoising steps, making the choice of sampling parameters crucial for minimizing discretization error. Non-asymptotic convergence bounds characterize sampling complexity, but their worst-case constants can obscure target geometry and thereby limit guidance on parameter optimization. Rather than bounding the error, we derive asymptotically exact small-stepsize expansions of Euler-Maruyama weak and Frechet errors for general smooth reverse diffusions, with explicit formulas for Gaussian data. These formulas provide tractable objectives for optimizing diffusion parameters, including the noise and rescaling schedules and the stochasticity coefficient, according to the target's covariance spectrum. In particular, our theory predicts lower optimal stochasticity at smaller step budgets, shows how to adapt the rescaling coefficient to the data power spectrum, and motivates a new family of effective noise schedules. A perturbative extension to Gaussian scale mixtures (GSMs) quantifies how departures from Gaussianity shift the optimal parameters. Finally, experiments on different real image datasets show that FID-optimal parameters agree with the qualitative theoretical predictions.

Diffusion models over discrete spaces have recently shown striking empirical success, yet their theoretical foundations remain incomplete. In this paper, we study the sampling efficiency of score-based discrete diffusion models under a continuous-time Markov chain (CTMC) formulation, with a focus on $τ$-leaping-based samplers. We establish sharp convergence guarantees for attaining $\varepsilon$ accuracy in Kullback-Leibler (KL) divergence for both uniform and masking noising processes. For uniform discrete diffusion, we show that the $τ$-leaping algorithm achieves an iteration complexity of order $\tilde O(d/\varepsilon)$, with $d$ the ambient dimension of the target distribution, eliminating linear dependence on the vocabulary size $S$ and improving existing bounds by a factor of $d$; moreover, we establish a matching algorithmic lower bound showing that linear dependence on the ambient dimension is unavoidable in general. For masking discrete diffusion, we introduce a modified $τ$-leaping sampler whose convergence rate is governed by an intrinsic information-theoretic quantity, termed the effective total correlation, which is bounded by $d \log S$ but can be sublinear or even constant for structured data. As a consequence, the sampler provably adapts to low-dimensional structure without prior knowledge or algorithmic modification, yielding sublinear convergence rates for various practical examples (such as hidden Markov models, image data, and random graphs). Our analysis requires no boundedness or smoothness assumptions on the score estimator beyond control of the score entropy loss.

Despite their empirical success, why diffusion models overcome the bottlenecks of classical score-based samplers remains unclear. In this work, we leverage Gaussian distributions to isolate this phenomenon. We establish 2-Wasserstein convergence bounds for optimized hyperparameters, showing that diffusion processes achieve a sampling error of $O(\sqrt{dλ_{\max}}\log N/N)$, where $d$ is the dimension, $N$ the number of sampling steps, and $λ_{\max}$ the largest eigenvalue of the target covariance matrix. Unadjusted and underdamped Langevin dynamics suffer from an additional $\sqrtκ$ factor, where $κ$ is the condition number. These rates follow from spectral bounds which are sharp: we confirm them via matching first-order asymptotics as $N\rightarrow\infty$. Our analysis provides a rigorous characterization, in the Gaussian setting, of how time-dependent score trajectories remove condition-number dependence during sampling. By contrast, in the learning phase, we show that estimating the unnoised score by gradient descent leads to essentially the same estimator as estimating a noisy score, which suggests that the benefits of noising do not come from the learning phase.