Convergence rate of Tsallis entropic regularized optimal transport

• URL: http://arxiv.org/abs/2304.06616v2
• Date: Mon, 07 Oct 2024 18:06:53 GMT
• Title: Convergence rate of Tsallis entropic regularized optimal transport
• Authors: Takeshi Suguro, Toshiaki Yachimura,
• Abstract summary: We establish fundamental results such as the $Gamma$-convergence of the Tsallis regularized optimal transport to the Monge--Kantorovich problem as the regularization parameter tends to zero. We show that the KL regularization achieves the fastest convergence rate in the Tsallis framework.
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• Abstract: In this paper, we study the Tsallis entropic regularized optimal transport in the continuous setting and establish fundamental results such as the $\Gamma$-convergence of the Tsallis regularized optimal transport to the Monge--Kantorovich problem as the regularization parameter tends to zero. In addition, using the quantization and shadow arguments developed by Eckstein--Nutz, we derive the convergence rate of the Tsallis entropic regularization and provide explicit constants. Furthermore, we compare these results with the well-known case of the Kullback--Leibler (KL) divergence regularization and show that the KL regularization achieves the fastest convergence rate in the Tsallis framework.

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