2022/09/07 by Amanjit Singh Kainth, Kainth, Amanjit Singh, Ting‐Kam Leonard Wong +3 · 1 citation
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2209.02938
openalex publication_date 2022/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The logarithmic divergence is an extension of the Bregman divergence motivated by optimal transport and a generalized convex duality, and satisfies many remarkable properties. Using the geometry induced by the logarithmic divergence, we introduce a generalization of continuous time mirror descent that we term the conformal mirror descent. We derive its dynamics under a generalized mirror map, and show that it is a time change of a corresponding Hessian gradient flow. We also prove convergence results in continuous time. We apply the conformal mirror descent to online estimation of a generalized exponential family, and construct a family of gradient flows on the unit simplex via the Dirichlet optimal transport problem.