通过凸化降维，实现高效得分扩散采样。
Fast Score-Based Sampling via Log-Concave Reductions
- 将复杂采样问题转化为一系列强对数凹子问题。
- 在维度d下达到误差ε时，复杂度为O(K√d polylog(1/ε))。
- 适用于高条件数分布，适合需要快速采样的生成模型研究者。
基于得分扩散的采样已取得显著的实证成果，受到多个研究领域的广泛关注。其依赖于不同噪声水平下的（近似）Stein得分函数。本文证明，在一般情况下，得分的存在性可将原问题“简化”为一系列自适应构造的K个强对数凹（SLC）子问题。该简化过程简单、可构造且与算法无关，因此任意SLC采样器均可作为子程序使用。由此直接导出多种得分采样复杂度界：例如，高精度SLC采样器可实现精度ε下O(K√d polylog(1/ε))的保证；随机中点方案则给出O(K d^{1/3} poly(1/ε))的界。当原始分布本身为SLC时，证明K ≤ 1 + log₂(κ)，首次获得对条件数κ呈对数依赖的高效过程；对一般分布，K依赖于路径上得分黑塞矩阵的几何结构。分析直接简洁，所用技巧与标准离散扩散分析互补。
原文摘要 · Abstract (English)
Sampling based on score diffusions has led to striking empirical results, and has attracted considerable attention from various research communities. It depends on availability of (approximate) Stein score functions for various levels of additive noise. We show how in some generality, the availability of scores allows the general problem to be ``reduced'' to sampling from an adaptively constructed sequence of $K$ strongly log-concave (SLC) sub-problems. The reduction is simple, constructive and algorithm-independent, so that any SLC sampler can be used as a subroutine. Various bounds on score-based sampling complexity follow directly: for instance, high-accuracy SLC samplers yield $\tilde{\mathcal{O}}(K \sqrt{d} \operatorname{polylog}(1/\varepsilon))$ guarantees for accuracy $\varepsilon$ in dimension $d$, where randomized midpoint SLC schemes yield $\tilde{\mathcal{O}}(K d^{1/3} \operatorname{poly}(1/\varepsilon))$ guarantees. When the original distribution itself is SLC, we prove that $K \leq 1 + \log_2(κ)$, thereby obtaining the first efficient procedure with logarithmic dependence on condition number $κ$; for general distributions, the quantity $K$ depends on the geometry of score Hessian across the trajectory. Our analysis is direct and simple, involving techniques and insights complementary to those in standard analyses of discretized diffusions.
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