2-格罗莫夫-瓦瑟斯坦距离的样本复杂度
Sample Complexity for the 2-Gromov-Wasserstein Distance
浏览论文内容
中文总结 AI 辅助
本文研究欧氏空间紧支撑概率测度间2-格罗莫夫-瓦瑟斯坦距离经验插件估计器的样本复杂度，证明其速率由两环境维度较小者决定，临界维度下除对数因子外最优。
中文摘要 AI 辅助
本文研究欧氏空间上紧支撑概率测度间2-格罗莫夫-瓦瑟斯坦距离$D_2$的经验插件估计器的样本复杂度。设$\boldsymbol{\text{μ}}$和$\boldsymbol{\text{ν}}$分别支撑在$\boldsymbol{\text{R}}^{d_x}$和$\boldsymbol{\text{R}}^{d_y}$的紧子集上，$\boldsymbol{\text{\text{μ}}}_n$和$\boldsymbol{\text{\text{ν}}}_n$是基于大小为$n$的独立样本得到的经验测度。我们证明：$\boldsymbol{\text{E}}\boldsymbol{|D_2^2(\boldsymbol{\text{\text{μ}}}_n,\boldsymbol{\text{\text{ν}}}_n)-D_2^2(\boldsymbol{\text{μ}},\boldsymbol{\text{ν}})\boldsymbol{| \boldsymbol{\text{≲}} n^{-2/((d_x\boldsymbol{\text{∧}} d_y)\boldsymbol{\text{∨}} 4)} (\boldsymbol{\text{log}} n)^{\boldsymbol{\text{1}}_{\boldsymbol{\text{\text{}}}\boldsymbol{\text{\text{∧}} d_y=4\boldsymbol{\text{}}}}}$，该速率在临界维度下除对数因子外是最优的。证明基于欧氏距离作为半空间特征映射的平方$\boldsymbol{L}^2$距离的几何表示，这给出了格罗莫夫-瓦瑟斯坦泛函的变分对偶形式，对应于由无限维辅助参数索引的一族经典最优传输问题。尽管所得代价函数在两个自变量中不一定是半凹的，我们引入代价的边际中心化以恢复最优度量-熵界所需的凹性结构。将该表示与经验过程估计结合，得到由两个环境维度中较小者决定的速率。
英文摘要
In this paper, we study the sample complexity of the empirical plug-in estimator for the $2$-Gromov-Wasserstein distance $D_2$ between compactly supported probability measures on Euclidean spaces. Let $μ$ and $ν$ be supported on compact subsets of $\mathbb{R}^{d_x}$ and $\mathbb{R}^{d_y}$, respectively, and let $\widehatμ_n$ and $\widehatν_n$ be their empirical measures based on independent samples of size $n$. We prove that \[ \mathbb{E}\left|D_2^2(\widehatμ_n,\widehatν_n)-D_2^2(μ,ν)\right| \lesssim n^{-2/((d_x\wedge d_y)\vee 4)} (\log n)^{\mathbf 1_{\{d_x\wedge d_y=4\}}}. \] This rate is sharp up to the logarithmic factor in the critical dimension. The proof is based on a geometric representation of the Euclidean distance as a squared $L^2$-distance between half-space feature maps. This yields a variational dual formulation of the Gromov-Wasserstein functional in terms of a family of classical optimal transport problems indexed by an infinite-dimensional auxiliary parameter. Although the resulting cost functions need not be semiconcave in either argument, we introduce a marginal recentering of the costs that restores the concavity structure needed for sharp metric-entropy bounds. Combining this representation with empirical-process estimates gives a rate governed by the smaller of the two ambient dimensions.
