Optimal transport: theory and algorithms

Optimal transport compares measures by the least cost of moving one onto the other. It requires the measures to have equal mass, and every unit of mass must be transported, so a small amount of mass far from the rest can dominate the comparison. Unbalanced optimal transport relaxes the marginal constraints and penalises the creation and destruction of mass alongside its transport. Its central example is the Hellinger–Kantorovich distance, a metric on nonnegative measures that combines Wasserstein-type transport with Hellinger-type change of mass.

Barycentres. Given measures $\mu_1, \dots, \mu_N$ and weights $\lambda_i$, a barycentre solves

In the Wasserstein setting this is equivalent to a multi-marginal problem with a least-cost formulation. We show that the Hellinger–Kantorovich barycentre admits a least-cost soft multi-marginal formulation, provided a one-sided hard marginal constraint is imposed, and that the constrained problem admits a conic multi-marginal reformulation with a single joint perspective cost function, in place of a separate two-marginal cost for each input. The constrained barycentre is a natural extension of the Wasserstein barycentre to the unbalanced setting.

Entropic regularisation. Adding an entropy term makes optimal transport problems strictly convex and solvable by Sinkhorn-type scaling algorithms. For unbalanced problems there are two qualitatively distinct regularisations, on the original space or on the extended space. We derive reformulations of both, introduce a regularised induced marginal perspective cost function, which gives an extended-space formulation of the original-space regularisation, and prove convergence to the unregularised problem for the extended-space regularisation.

Semi-discrete optimal transport, between a continuous measure and a discrete one, is also the computational core of our work on modelling microstructure.