Kd-tree Based Wasserstein Distance Approximation for High-Dimensional Data

TL;DR

This paper introduces kd-Flowtree, a kd-tree-based method for approximating Wasserstein distance in high-dimensional data, achieving better accuracy and faster computation than existing methods for large-scale retrieval tasks.

Contribution

The paper presents kd-Flowtree, a novel kd-tree-based approximation method that maintains high accuracy and reduces preprocessing time in high-dimensional Wasserstein distance computations.

Findings

kd-Flowtree outperforms existing methods in retrieval accuracy.

It provides a dataset-size-independent probabilistic accuracy bound.

kd-Flowtree reduces preprocessing and computation time.

Abstract

The Wasserstein distance is a discrepancy measure between probability distributions, defined by an optimal transport problem. It has been used for various tasks such as retrieving similar items in high-dimensional images or text data. In retrieval applications, however, the Wasserstein distance is calculated repeatedly, and its cubic time complexity with respect to input size renders it unsuitable for large-scale datasets. Recently, tree-based approximation methods have been proposed to address this bottleneck. For example, the Flowtree algorithm computes transport on a quadtree and evaluates cost using the ground metric, and clustering-tree approaches have been reported to achieve high accuracy. However, these existing trees often incur significant construction time for preprocessing, and crucially, standard quadtrees cannot grow deep enough in high-dimensional spaces, resulting in poor approximation accuracy. In this paper, we propose kd-Flowtree, a kd-tree-based Wasserstein distance approximation method that uses a kd-tree for data embedding. Since kd-trees can grow sufficiently deep and adaptively even in high-dimensional cases, kd-Flowtree is capable of maintaining good approximation accuracy for such cases. In addition, kd-trees can be constructed quickly than quadtrees, which contributes to reducing the computation time required for nearest neighbor search, including preprocessing. We provide a probabilistic upper bound on the nearest-neighbor search accuracy of kd-Flowtree, and show that this bound is independent of the dataset size. In the numerical experiments, we demonstrated that kd-Flowtree outperformed the existing Wasserstein distance approximation methods for retrieval tasks with real-world data.