Distribution comparison plays a central role in many machine learning tasks like data classification and generative modeling. In this study, we propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance between two probability distributions with low complexity. In particular, we first project two high-dimensional probability distributions using Hilbert curve to obtain a coupling between them, and then calculate the transport distance between these two distributions in the original space, according to the coupling. We show that HCP distance is a proper metric and is well-defined for probability measures with bounded supports. Furthermore, we demonstrate that the modified empirical HCP distance with the $L_p$ cost in the $d$-dimensional space converges to its population counterpart at a rate of no more than $O(n^{-1/2\max\{d,p\}})$. To suppress the curse-of-dimensionality, we also develop two variants of the HCP distance using (learnable) subspace projections. Experiments on both synthetic and real-world data show that our HCP distance works as an effective surrogate of the Wasserstein distance with low complexity and overcomes the drawbacks of the sliced Wasserstein distance.

翻译：分布比较在许多机器学习任务（如数据分类和生成建模）中发挥着核心作用。本研究提出一种新型度量——希尔伯特曲线投影（HCP）距离，用于以低复杂度衡量两个概率分布之间的距离。具体而言，我们首先利用希尔伯特曲线对两个高维概率分布进行投影以获得其耦合关系，然后根据该耦合计算原始空间中这两个分布之间的传输距离。我们证明HCP距离是一种合适的度量，且对具有有界支撑的概率测度定义良好。此外，我们证明在d维空间中采用$L_p$代价的修正经验HCP距离以不超过$O(n^{-1/2\max\{d,p\}})$的速率收敛至其总体对应量。为抑制维数灾难，我们还利用（可学习的）子空间投影开发了HCP距离的两种变体。在合成数据和真实数据上的实验表明，我们的HCP距离可作为Wasserstein距离的有效替代，兼具低复杂度并克服了切片Wasserstein距离的缺陷。