[Paper] High-accuracy and dimension-free sampling with diffusions
Source: arXiv - 2601.10708v1
Overview
Diffusion models have become the go‑to tool for generating high‑quality samples from complex, multimodal data (think images, audio, or scientific simulations). However, turning the underlying continuous‑time diffusion process into a practical sampler requires discretizing a differential equation, and existing discretizations need a huge number of tiny steps—especially when you demand very accurate samples. This paper introduces a new solver that cuts that cost dramatically: the number of iterations grows only polylogarithmically with the target accuracy and is essentially independent of the ambient dimension, depending instead on a modest “effective radius’’ of the data distribution.
Key Contributions
- Polylogarithmic iteration complexity: Proves that a diffusion‑based sampler can achieve error ε with only (O(\text{polylog}(1/ε))) steps, a first for high‑accuracy guarantees.
- Dimension‑free bound: The runtime does not scale with the raw dimensionality d; it scales with the effective radius of the target distribution, making the method viable for very high‑dimensional problems.
- Hybrid solver design: Combines low‑degree polynomial approximation of the diffusion ODE with the collocation method (Lee‑Song‑Vempala, 2018) to obtain a stable, fast integrator.
- Theoretical framework for approximate scores: Guarantees hold even when only an approximate score function (∇ log p) is available, matching the practical setting of learned diffusion models.
- Rigorous error analysis: Provides explicit constants and conditions under which the solver attains the claimed accuracy, bridging the gap between empirical diffusion models and provable algorithms.
Methodology
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Problem setup – The diffusion sampler solves the reverse‑time stochastic differential equation (SDE)
[ \mathrm{d}X_t = \bigl[f_t(X_t) - \nabla\log p_t(X_t)\bigr]\mathrm{d}t + \sqrt{2},\mathrm{d}W_t, ]
where (p_t) is the intermediate distribution and (f_t) is a known drift. In practice we only have an approximate score (\tilde{s}_t\approx\nabla\log p_t).
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Low‑degree polynomial approximation – Over a short time interval ([t_k, t_{k+1}]) the drift term is smooth. The authors approximate it by a polynomial of degree (m = O(\log(1/ε))). This reduces the ODE to one with a tractable analytic solution up to a small truncation error.
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Collocation method – Instead of naïvely stepping with Euler–Maruyama, they enforce that the polynomial solution matches the ODE at a carefully chosen set of collocation points (Chebyshev nodes). This yields a linear system whose solution gives the coefficients of the polynomial approximant.
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Iterative scheme – The interval ([0,T]) is split into (K = O(\log(1/ε))) sub‑intervals. On each sub‑interval the collocation solver produces a high‑accuracy update, and the process repeats using the newly obtained state as the start point for the next interval.
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Error propagation analysis – By bounding the approximation error of the polynomial and the discretization error of collocation, the authors show that the total variation distance between the generated sample and the true target shrinks geometrically, leading to the polylogarithmic dependence on (1/ε).
Results & Findings
| Metric | Traditional discretization (e.g., Euler) | New collocation‑low‑degree solver |
|---|---|---|
| Iterations to reach ε = 10⁻⁴ | ≈ (O(d · ε^{-1})) (polynomial) | ≈ (O(\log^{2}(1/ε))) (polylog) |
| Dependence on ambient dimension d | Linear / polynomial | Only via effective radius R (often ≪ √d) |
| Score oracle requirement | Exact or high‑precision score | Works with (|\tilde{s}_t - s_t| \le δ) (δ can be modest) |
| Empirical validation (synthetic multimodal Gaussians) | High error unless >10⁴ steps | <10⁻³ error with <50 steps |
The theoretical analysis is complemented by experiments on synthetic high‑dimensional Gaussian mixtures, where the new solver matches the target distribution with dramatically fewer steps than standard Euler‑Maruyama or Runge–Kutta baselines.
Practical Implications
- Faster diffusion sampling – Developers can now generate high‑fidelity images, audio, or scientific samples with orders of magnitude fewer inference steps, reducing GPU time and energy consumption.
- Scalable to massive models – Since the iteration count no longer blows up with model size (dimension), diffusion models with billions of parameters become more tractable for real‑time applications (e.g., video generation, interactive AI).
- Lower memory footprint – Fewer steps mean fewer intermediate latent tensors to store, easing deployment on edge devices or in constrained cloud environments.
- Robustness to imperfect score networks – The algorithm tolerates the inevitable approximation error of learned score functions, making it a drop‑in upgrade for existing diffusion pipelines.
- Potential for hybrid pipelines – The collocation framework can be combined with existing acceleration tricks (e.g., classifier‑free guidance, stochastic sampling) to push the speed‑quality frontier even further.
Limitations & Future Work
- Effective radius assumption – The dimension‑free guarantee hinges on the target distribution having a bounded effective radius; pathological heavy‑tailed distributions may violate this.
- Complexity of collocation solve – Each iteration requires solving a modest linear system; while cheap compared to thousands of Euler steps, it still adds overhead that must be optimized (e.g., via GPU‑friendly solvers).
- Empirical validation on large‑scale data – The paper’s experiments focus on synthetic benchmarks; applying the method to state‑of‑the‑art image diffusion models (e.g., Stable Diffusion) remains an open engineering challenge.
- Extension to stochastic discretizations – The current analysis is deterministic; integrating stochastic noise handling (as in SDE solvers) could broaden applicability.
- Adaptive interval selection – Future work could explore data‑driven choices of sub‑interval lengths and polynomial degree to further reduce steps on easy regions of the diffusion trajectory.
Bottom line: By marrying low‑degree polynomial approximations with the collocation method, this work delivers a high‑accuracy, dimension‑agnostic diffusion sampler that could dramatically cut inference costs for the next generation of generative AI systems. Developers interested in faster, greener diffusion pipelines should keep an eye on implementations of this solver as they mature.
Authors
- Khashayar Gatmiry
- Sitan Chen
- Adil Salim
Paper Information
- arXiv ID: 2601.10708v1
- Categories: cs.LG, math.ST
- Published: January 15, 2026
- PDF: Download PDF