2019/06/17 by Ting‐Kam Leonard Wong, Wong, Ting-Kam Leonard, Jiaowen Yang +1
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Mechanics and Entropy #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1906.09103
openalex publication_date 2019/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the logarithmic L(α)-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(α)-divergence is a canonical divergence of a statistical manifold with constant sectional curvature -α. 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.