arXiv subjects

Jiaowen Yang

Publications and source records attributed to Jiaowen Yang.

3 recordsLinked to original sources

arXiv2025

Maximum likelihood estimation for the $λ$-exponential family

The $λ$-exponential family generalizes the standard exponential family via a generalized convex duality motivated by optimal transport. It is the constant-curvature analogue of the exponential family from the information-geometric point of view, but the development of computational methodologies is still in an early stage. In this paper, we propose a fixed point iteration for maximum likelihood estimation under i.i.d.~sampling, and prove using the duality that the likelihood is monotone along the iterations. We illustrate the algorithm with the $q$-Gaussian distribution and the Dirichlet perturbation.

math.ST↗

arXiv2021

Pseudo-Riemannian geometry embeds information geometry in optimal transport

Optimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate-invariant properties of statistical inference. Their connections and applications in statistics and machine learning have started to gain more attention. In this paper we give a new differential geometric connection between the two fields. Namely, the pseudo-Riemannian framework of Kim and McCann, a geometric perspective on the fundamental Ma-Trudinger-Wang (MTW) condition in the regularity theory of optimal transport maps, encodes the dualistic structure of statistical manifold. This general relation is described using the natural framework of $c$-divergence, a divergence defined by an optimal transport map. As a by-product, we obtain a new information-geometric interpretation of the MTW tensor. This connection sheds light on old and new aspects of information geometry. The dually flat geometry of Bregman divergence corresponds to the quadratic cost and the pseudo-Euclidean space, and the $L^{(α)}$-divergence introduced by Pal and the first author has constant sectional curvature in a sense to be made precise. In these cases we give a geometric interpretation of the information-geometric curvature in terms of the divergence between a primal-dual pair of geodesics.

math.DG↗

arXiv2019

Logarithmic divergences: geometry and interpretation of curvature

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.

math.DG↗