Optimal transport describes the most efficient way to move mass between two distributions, given a cost matrix for moving mass between each pair of locations. Entropic optimal transport, solved via the Sinkhorn algorithm, is a widely used regularized version of this problem. Its inverse problem asks the opposite question: given an observed transport plan, what cost matrix produced it? This is difficult because the cost is identifiable only up to an additive gauge freedom. Here we show that this freedom can be fixed exactly by a single double-centering operation applied to the observed plan, yielding the true cost matrix in closed form, with no iterative optimization required. When a modest number of true cost entries are known, the same approach lets us jointly estimate the temperature parameter controlling the entropic regularization, together with a diagnostic for the reliability of this estimate. We further show that the method is not specific to the entropic optimal transport, but extends to a broader class of transport models defined by an invertible relation between cost and plan.
Optimal transport asks for the cheapest way of rearranging one distribution of mass into another, and the answer supplies a metric on probability measures that respects the geometry of the underlying space. This review traces the subject from the formulation posed by Monge in 1781, through the linear relaxation introdu...
Assanu Augustine, Assanu Augustine· International Journal of Pur...· 0 citations
We study the gap between the integrated direction-wise kinetic minima and the minimum expected Dirichlet energy of a joint particle realization of a prescribed two-parameter family of conditional probability measures. Under regular coarea assumptions with compact connected fibres and a star-shaped parameter domain, we...
We prove quantitative stability estimates for quadratic optimal transport maps by regularizing a Brenier potential and estimating the error in the duality gap. The source has an $L^p$ density, $p>1$, relative to a probability measure with a BV density. When one target lies in a fixed ball, we obtain the $W_2$ H\"older...
We study quadratically regularized optimal transport with quadratic cost in the regime of small regularization $\varepsilon$, where the support of the optimal coupling is sparse. For smooth marginal densities in $\mathbb{R}^d$, we show that the support contains the graph of the Brenier map. After centering by the Breni...
The problem of incompressible fluid mixing is long-studied, yet many questions remain open. This paper aims to address the question ``what do efficient flow fields for mixing look like, and how do they behave?''Inspired by the dynamic and geometric approach to optimal mass transport, we formulate a version of the fluid...
Max Emerick, John Igraszek, Bassam Bamieh· 0 citations
A central question in data-driven inverse problems is how to construct a regularizer that adapts to the geometry of the data distribution. Recent work in optimal regularization shows that, within a broad Gibbs class, the regularizer best matched to a distribution $P$ is determined by a single, direction-dependent radia...
Carson Newman, Oscar Leong· 0 citations
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