提出流模型在Wasserstein度量下的$O( oot d floor)$采样复杂度最优界。
Pathway to $O(\sqrt{d})$ Complexity bound under Wasserstein metric of flow-based models
- 通过后向流映射Lipschitz性与局部离散误差分析误差来源。
- 在高斯尾部假设下，采样复杂度随$ oot d floor$线性增长。
- 适用于Föllmer过程和1-修正流模型，理论指导设计新算法。
我们提供了估计流生成模型在Wasserstein度量下误差的可实现解析工具，并建立了关于维度的最优采样迭代复杂度界$O( oot d floor)$。误差可明确分解为两部分：后向流的推送映射Lipschitz性，其与维度无关；以及局部离散误差，随维度呈$O( oot d floor)$增长。前者源于热流诱导的Lipschitz变量变换存在性，后者依赖于得分函数在空间和时间方向的正则性。这些假设在关联Föllmer过程与1-修正流、满足高斯尾部假设的流生成模型中成立。因此，我们证明了采样迭代复杂度与协方差算子迹的平方根成线性关系，该迹与前向过程的不变分布相关。
原文摘要 · Abstract (English)
We provide attainable analytical tools to estimate the error of flow-based generative models under the Wasserstein metric and to establish the optimal sampling iteration complexity bound with respect to dimension as $O(\sqrt{d})$. We show this error can be explicitly controlled by two parts: the Lipschitzness of the push-forward maps of the backward flow which scales independently of the dimension; and a local discretization error scales $O(\sqrt{d})$ in terms of dimension. The former one is related to the existence of Lipschitz changes of variables induced by the (heat) flow. The latter one consists of the regularity of the score function in both spatial and temporal directions. These assumptions are valid in the flow-based generative model associated with the Föllmer process and $1$-rectified flow under the Gaussian tail assumption. As a consequence, we show that the sampling iteration complexity grows linearly with the square root of the trace of the covariance operator, which is related to the invariant distribution of the forward process.
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