Marie-Therese Wolfram

Inverse Optimal Transport

Discrete optimal transportation problems arise in various contexts in engineering, the sciences and the social sciences. Examples include the marriage market in economics or international migration flows in demographics. Often the underlying cost criterion is unknown, or only partly known, and the observed optimal solutions are corrupted by noise. In this talk we discuss a systematic approach to infer unknown costs from noisy observations of optimal transportation plans. The proposed methodologies are developed within the Bayesian framework for inverse problems and require only the ability to solve the forward optimal transport problem, which is a linear program, and to generate random numbers. We illustrate our approach using the example of international migration flows. Here reported migration flow data captures (noisily) the number of individuals moving from one country to another in a given period of time. It can be interpreted as a noisy observation of an optimal transportation map, with costs related to the geographical position of countries. We use a graph-based formulation of the problem, with countries at the nodes of graphs and non-zero weighted adjacencies only on edges between countries which share a border. We use the proposed algorithm to estimate the weights, which represent cost of transition, and to quantify uncertainty in these weights.