Kantorovich Spaces and Optimization

A generalized dual maximizer for the Monge–Kantorovich transport problem

The dual attainment of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X,Y are assumed to be polish and equipped with Borel probability measures μ and ν. The transport cost function c : X × Y → [0,∞] is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive meas...

Convexity conditions of Kantorovich function and related semi-infinite linear matrix inequalities

The Kantorovich function (xT Ax)(xT A−1x), where A is a positive definite matrix, is not convex in general. From a matrix or convex analysis point of view, it is interesting to address the question: When is this function convex? In this paper, we prove that the 2dimensional Kantorovich function is convex if and only if the condition number of its matrix is less than or equal to 3 + 2 √ 2. Thus ...

A Probability Monad as the Colimit of Finite Powers

We define a monad on the category of complete metric spaces with short maps, which assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich–Wasserstein distance. It is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1bounded complete metric spaces...

Imaging with Kantorovich-Rubinstein Discrepancy

We propose the use of the Kantorovich-Rubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularisation model endowed with a Kantorovich-Rubinstein discrepancy term and total variation regularization in the context of image denoising and cartoon-texture decomposition. We point out connections of this approach to several other recently proposed me...

Research Article On the Generalized Favard-Kantorovich and Favard-Durrmeyer Operators in Exponential Function Spaces