固定支集Wasserstein重心：计算困难与快速算法
Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm
摘要：
我们研究固定支持Wasserstein重心问题（FS-WBP），它包括在计算的Wasserstein重心的$m$离散概率测度上支持一个有限的度量空间的大小$n$。我们首先表明，从标准的线性规划（LP）表示的FS-WBP的约束矩阵是\textit{not totally unimodular}时，$m \geq 3$和$n \geq 3$。这一结果解决了一个悬而未决的问题有关的FS-WBP和最小费用流（MCF）问题之间的关系，因为它证明了在标准LP形式的FS-WBP是不是一个MCF问题时，$m \geq 3$和$n \geq 3$。我们还开发了一个可证明的快速\textit{deterministic}变体的著名的迭代Bregman投影（IBP）算法，命名为\textsc{FastIBP}，与复杂性界的$\tilde{O}(mn^{7/3}\varepsilon^{-4/3})$，其中$\varepsilon \in (0, 1)$是所需的公差。该复杂性界优于已知的IBP算法的$\tilde{O}(mn^2\varepsilon^{-2})$的$\varepsilon$复杂性界，以及加速交替极小化算法或加速原始-对偶自适应梯度算法的$\tilde{O}(mn^{5/2}\varepsilon^{-1})$的$n$复杂性界.最后，我们进行了大量的实验与合成数据和真实图像和证明良好的性能的\textsc{FastIBP}算法在实践中。
英文摘要：
We study the fixed-support Wasserstein barycenter problem (FS-WBP), which consists in computing the Wasserstein barycenter of $m$ discrete probability measures supported on a finite metric space of size $n$. We show first that the constraint matrix arising from the standard linear programming (LP) representation of the FS-WBP is \textit{not totally unimodular} when $m \geq 3$ and $n \geq 3$. This result resolves an open question pertaining to the relationship between the FS-WBP and the minimum-cost flow (MCF) problem since it proves that the FS-WBP in the standard LP form is not an MCF problem when $m \geq 3$ and $n \geq 3$. We also develop a provably fast \textit{deterministic} variant of the celebrated iterative Bregman projection (IBP) algorithm, named \textsc{FastIBP}, with a complexity bound of $\tilde{O}(mn^{7/3}\varepsilon^{-4/3})$, where $\varepsilon \in (0, 1)$ is the desired tolerance. This complexity bound is better than the best known complexity bound of $\tilde{O}(mn^2\varepsilon^{-2})$ for the IBP algorithm in terms of $\varepsilon$, and that of $\tilde{O}(mn^{5/2}\varepsilon^{-1})$ from accelerated alternating minimization algorithm or accelerated primal-dual adaptive gradient algorithm in terms of $n$. Finally, we conduct extensive experiments with both synthetic data and real images and demonstrate the favorable performance of the \textsc{FastIBP} algorithm in practice.
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