Fixed Support Tree-Sliced Wasserstein Barycenter

• URL: http://arxiv.org/abs/2109.03431v1
• Date: Wed, 8 Sep 2021 04:50:33 GMT
• Title: Fixed Support Tree-Sliced Wasserstein Barycenter
• Authors: Yuki Takezawa, Ryoma Sato, Zornitsa Kozareva, Sujith Ravi, Makoto Yamada
• Abstract summary: We propose a barycenter under the tree-Wasserstein distance, called the fixed support tree-Wasserstein barycenter (FS-TWB) and its extension, called the fixed support tree-sliced Wasserstein barycenter (FS-TSWB) We show that the FS-TWB and FS-TSWB can be solved two orders of magnitude faster than the original Wasserstein barycenter.
• Score: 40.087553363884396
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Wasserstein barycenter has been widely studied in various fields, including natural language processing, and computer vision. However, it requires a high computational cost to solve the Wasserstein barycenter problem because the computation of the Wasserstein distance requires a quadratic time with respect to the number of supports. By contrast, the Wasserstein distance on a tree, called the tree-Wasserstein distance, can be computed in linear time and allows for the fast comparison of a large number of distributions. In this study, we propose a barycenter under the tree-Wasserstein distance, called the fixed support tree-Wasserstein barycenter (FS-TWB) and its extension, called the fixed support tree-sliced Wasserstein barycenter (FS-TSWB). More specifically, we first show that the FS-TWB and FS-TSWB problems are convex optimization problems and can be solved by using the projected subgradient descent. Moreover, we propose a more efficient algorithm to compute the subgradient and objective function value by using the properties of tree-Wasserstein barycenter problems. Through real-world experiments, we show that, by using the proposed algorithm, the FS-TWB and FS-TSWB can be solved two orders of magnitude faster than the original Wasserstein barycenter.

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