# Optimal Transport on the Probability Simplex with Logarithmic Cost

@article{Khan2018OptimalTO, title={Optimal Transport on the Probability Simplex with Logarithmic Cost}, author={Gabriel Khan and Jun Zhang}, journal={arXiv: Optimization and Control}, year={2018} }

Motivated by the financial problem of building financial portfolios which outperform the market, Pal and Wong considered optimal transport on the probability simplex $\triangle^n$ where the cost function is induced by the free energy. We study the regularity of this problem and find that the associated $MTW$ tensor is non-negative definite and in fact constant on $\triangle^n \times \triangle^n$. We further find that relative $c$-convexity corresponds to the standard notion of convexity in the… Expand

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#### References

SHOWING 1-10 OF 54 REFERENCES

Logarithmic divergences from optimal transport and Rényi geometry

- Mathematics, Computer Science
- Information Geometry
- 2018

It is shown that if a statistical manifold is dually projectively flat with constant curvature, then it is locally induced by an L(∓α)-divergence, and a generalized Pythagorean theorem holds true. Expand

Hessian Curvature and Optimal Transport

- Computer Science, Mathematics
- GSI
- 2019

A new complex geometric interpretation of the optimal transport problem is introduced by considering an induced Sasaki metric on the tangent bundle of the domain of \(\varPsi) and the Ma-Trudinger-Wang tensor is proportional to the orthogonal bisectional curvature. Expand

On the regularity of solutions of optimal transportation problems

- Mathematics
- 2009

We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition… Expand

Multiplicative Schrödinger problem and the Dirichlet transport

- Mathematics
- 2018

We consider an optimal transport problem on the unit simplex whose solutions are given by gradients of exponentially concave functions and prove two main results. First, we show that the optimal… Expand

About the analogy between optimal transport and minimal entropy

- Mathematics
- 2015

We describe some analogy between optimal transport and the Schr\"odinger problem where the transport cost is replaced by an entropic cost with a reference path measure. A dual Kantorovich type… Expand

Hölder Continuity and Injectivity of Optimal Maps

- Mathematics
- 2011

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x, y).… Expand

Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 2

- Mathematics
- 2009

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to… Expand

A Jacobian Inequality for Gradient Maps on the Sphere and Its Application to Directional Statistics

- Mathematics
- 2009

In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this article, the Jacobian determinant of a gradient map… Expand

From the Schr\"odinger problem to the Monge-Kantorovich problem

- Mathematics
- 2010

The aim of this article is to show that the Monge-Kantorovich problem is the limit of a sequence of entropy minimization problems when a fluctuation parameter tends down to zero. We prove the… Expand

Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures

- Mathematics
- 2015

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by… Expand