Kd-tree Based Wasserstein Distance Approximation for High-Dimensional Data
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911384675024896 |
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| author | Teshigawara, Kanata Oh, Keisho Kobayashi, Ken Nakata, Kazuhide |
| author_facet | Teshigawara, Kanata Oh, Keisho Kobayashi, Ken Nakata, Kazuhide |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12975 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kd-tree Based Wasserstein Distance Approximation for High-Dimensional Data Teshigawara, Kanata Oh, Keisho Kobayashi, Ken Nakata, Kazuhide Data Structures and Algorithms 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. |
| title | Kd-tree Based Wasserstein Distance Approximation for High-Dimensional Data |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2601.12975 |