High-accuracy and dimension-free sampling with diffusions

2026-01-15

AI Summary

Diffusion models typically require numerous small-step iterations for high-accuracy sampling, and existing solvers exhibit iteration complexity that grows polynomially with both dimensionality and desired precision. This work proposes a novel solver that synergistically combines low-order function approximation with collocation methods, yielding the first diffusion sampler that relies solely on approximate access to the score of the data distribution. By leveraging the effective radius of the support set of the target distribution to control dimensional effects, the method achieves—through theoretical analysis—an iteration complexity that scales only polylogarithmically with the inverse of the accuracy and is independent of the ambient dimension. This represents a significant breakthrough over conventional samplers, which are fundamentally constrained by their joint dependence on dimensionality and precision.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Diffusion models have shown remarkable empirical success in sampling from rich multi-modal distributions. Their inference relies on numerically solving a certain differential equation. This differential equation cannot be solved in closed form, and its resolution via discretization typically requires many small iterations to produce \emph{high-quality} samples. More precisely, prior works have shown that the iteration complexity of discretization methods for diffusion models scales polynomially in the ambient dimension and the inverse accuracy $1/\varepsilon$. In this work, we propose a new solver for diffusion models relying on a subtle interplay between low-degree approximation and the collocation method (Lee, Song, Vempala 2018), and we prove that its iteration complexity scales \emph{polylogarithmically} in $1/\varepsilon$, yielding the first ``high-accuracy''guarantee for a diffusion-based sampler that only uses (approximate) access to the scores of the data distribution. In addition, our bound does not depend explicitly on the ambient dimension; more precisely, the dimension affects the complexity of our solver through the \emph{effective radius} of the support of the target distribution only.

Problem

• diffusion models
• sampling
• iteration complexity
• high-accuracy
• dimension-free

Innovation

• diffusion models
• high-accuracy sampling
• polylogarithmic complexity
• dimension-free
• collocation method