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Optimal transportation plans with escort entropy regularization

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Abstract

The entropy-regularized Wasserstein transportation problem is useful for solving lots of problems in various applications. We study the deformed escort entropy regularization including the q-escort regularization and prove that the family of the inverse escort distributions of the optimal transportation plans forms a deformed exponential family, which has dually flat information-geometric structure. This elucidates the role of the escort transformation and its inverse in the theory of deformed exponential families. We further prove that the regularized cost function gives the dual potential of the flat manifold. We derive a new divergence function between two probability distributions in a probability simplex and a related Riemannian metric based on the regularized cost.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1, Sanda, 669-1337, Japan

    Takashi Kurose

  2. Advanced Research Department, Frontier Research Center, Toyota Motor Corporation, 1, Toyota-cho, Aichi, 471-8572, Japan

    Shintaro Yoshizawa

  3. Teikyo University, ACRO, Tokyo, Japan

    Shun-ichi Amari

  4. RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

    Shun-ichi Amari

Authors
  1. Takashi Kurose
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  2. Shintaro Yoshizawa
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  3. Shun-ichi Amari
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Correspondence to Shun-ichi Amari.

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Kurose, T., Yoshizawa, S. & Amari, Si. Optimal transportation plans with escort entropy regularization. Info. Geo. 5, 79–95 (2022). https://doi.org/10.1007/s41884-021-00058-2

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  • DOI: https://doi.org/10.1007/s41884-021-00058-2

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