Hilbert Curve Projection Distance for Distribution Comparison

• URL: http://arxiv.org/abs/2205.15059v4
• Date: Tue, 6 Feb 2024 10:58:38 GMT
• Title: Hilbert Curve Projection Distance for Distribution Comparison
• Authors: Tao Li, Cheng Meng, Hongteng Xu, Jun Yu
• Abstract summary: We propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance between two probability distributions. We show that HCP distance is a proper metric and is well-defined for probability measures with bounded supports. 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.
• Score: 34.8765820950517
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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