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Logarithmic Divergences: Geometry and Interpretation of Curvature

Part of the Lecture Notes in Computer Science book series (LNIP,volume 11712)

Abstract

We study the logarithmic \(L^{(\alpha )}\)-divergence which extrapolates the Bregman divergence and corresponds to solutions to novel optimal transport problems. We show that this logarithmic divergence is equivalent to a conformal transformation of the Bregman divergence, and, via an explicit affine immersion, is equivalent to Kurose’s geometric divergence. In particular, the \(L^{(\alpha )}\)-divergence is a canonical divergence of a statistical manifold with constant sectional curvature \(-\alpha \). For such a manifold, we give a geometric interpretation of its sectional curvature in terms of how the divergence between a pair of primal and dual geodesics differ from the dually flat case. Further results can be found in our follow-up paper [27] which uncovers a novel relation between optimal transport and information geometry.

Keywords

  • Logarithmic divergence
  • Bregman divergence
  • Conformal divergence
  • Affine immersion
  • Constant sectional curvature
  • Optimal transport

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Correspondence to Ting-Kam Leonard Wong .

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Wong, TK.L., Yang, J. (2019). Logarithmic Divergences: Geometry and Interpretation of Curvature. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_43

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  • DOI: https://doi.org/10.1007/978-3-030-26980-7_43

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