Optimal transport has been very successful for various machine learning tasks; however, it is known to suffer from the curse of dimensionality. Hence, dimensionality reduction is desirable when applied to high-dimensional data with low-dimensional structures. The kernel max-sliced (KMS) Wasserstein distance is developed for this purpose by finding an optimal nonlinear mapping that reduces data into $1$ dimensions before computing the Wasserstein distance. However, its theoretical properties have not yet been fully developed. In this paper, we provide sharp finite-sample guarantees under milder technical assumptions compared with state-of-the-art for the KMS $p$-Wasserstein distance between two empirical distributions with $n$ samples for general $p\in[1,\infty)$. Algorithm-wise, we show that computing the KMS $2$-Wasserstein distance is NP-hard, and then we further propose a semidefinite relaxation (SDR) formulation (which can be solved efficiently in polynomial time) and provide a relaxation gap for the SDP solution. We provide numerical examples to demonstrate the good performance of our scheme for high-dimensional two-sample testing.

翻译：最优传输在各种机器学习任务中取得了显著成功；然而，它已知会受到维度诅咒的影响。因此，当应用于具有低维结构的高维数据时，降维是理想的选择。核最大切片（KMS）Wasserstein距离正是为此目的而发展起来的，它通过寻找一个将数据降至$1$维的最优非线性映射，然后再计算Wasserstein距离。然而，其理论性质尚未得到充分发展。在本文中，我们为两个具有$n$个样本的经验分布之间的KMS $p$-Wasserstein距离（对于一般的$p\in[1,\infty)$）提供了尖锐的有限样本保证，并且所需的技术假设比现有技术更为温和。在算法方面，我们证明了计算KMS $2$-Wasserstein距离是NP难的，随后我们进一步提出了一个半定松弛（SDR）公式（该公式可在多项式时间内高效求解），并为该SDP解提供了松弛间隙。我们提供了数值示例，以展示我们方案在高维双样本检验中的良好性能。