This conversion has been funded by the ERC project SIGMA-Vision, and it has been performed by Pierre Stock, congratulation for this nice work!

]]>- In the first paper “An Interpolating Distance between Optimal Transport and Fisher-Rao”, we proposed (simultaneously and independently with two other groups of researchers) a new geodesic distance between two arbitrary positive measures, generalizing OT to the unblanced cases (i.e. when the measures are not normalized to unit mass).
- In the second paper “Unbalanced Optimal Transport: Geometry and Kantorovich Formulation”, we showed (simultaneously and independently with another group of researchers) that this distance is a special case of a generic class of “static” OT problems, which generalizes the linear programming formulation of OT.
- In this last paper, “Scaling Algorithms for Unbalanced Transport Problems”, we show how to solve efficiently these problems, and many more (in particular also barycenters and gradient flows) using entropic regularization, a recent computational technic championed by Marco Cuturi. The resulting algorithm is a far reaching generalization of Sinkhorn iterative scaling method.

With Nicolas and Marco we propose in Wasserstein Barycentric Coordinates: Histogram Regression Using Optimal Transport a way to compute barycentric coordinates for histograms, according to the geometry defined by optimal transport (the so-called Wasserstein distance. It is useful to perform vizualization and to navigate in collections of histograms, and also enables to compute “projections” on a geodesic simplex defined by these histograms. The main contribution is a computationally tractable optimization scheme, that makes use of recursive differentiation of Sinkhorn’s algorithm.

With Justin, Vova and Suvrit, we propose in Entropic Metric Alignment for Correspondence Problems a fast iterative scaling algorithm to approximate the solution of quadratic assignment problems. It iteratively solves an entropic regularization of optimal transport, which in turn can be solved using Sinkhorn’s algorithm, and is similar to the “softassign” algorithm.

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