证明扩散模型在低维流形上收敛速度与内在维度线性相关
Linear Convergence of Diffusion Models Under the Manifold Hypothesis
- 基于流形假设，设计新积分方案提升收敛效率
- 收敛步数仅随内在维度d线性增长（含对数项）
- 适用于高维数据生成，尤其关注收敛速度的研究者
得分匹配生成模型在从复杂高维数据分布中采样方面表现优异。许多应用中，该分布被认为集中在嵌入于D维空间的低维d维流形上，这被称为流形假设。现有最优收敛保证要么与D线性相关，要么与d多项式（超线性）相关。后者利用了后向SDE的新积分方案。本文结合两者优势，证明扩散模型在KL散度意义下的收敛所需步数在内在维度d上为线性（含对数项）。此外，我们证明该线性依赖关系是紧的。
原文摘要 · Abstract (English)
Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower $d$-dimensional manifold embedded into $D$-dimensional space; this is known as the manifold hypothesis. The current best-known convergence guarantees are either linear in $D$ or polynomial (superlinear) in $d$. The latter exploits a novel integration scheme for the backward SDE. We take the best of both worlds and show that the number of steps diffusion models require in order to converge in Kullback-Leibler~(KL) divergence is linear (up to logarithmic terms) in the intrinsic dimension $d$. Moreover, we show that this linear dependency is sharp.
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