5 research outputs found
The GenCol algorithm for high-dimensional optimal transport: general formulation and application to barycenters and Wasserstein splines
We extend the recently introduced genetic column generation algorithm for
high-dimensional multi-marginal optimal transport from symmetric to general
problems. We use the algorithm to calculate accurate mesh-free Wasserstein
barycenters and cubic Wasserstein splines
The structure and normalized volume of Monge polytopes
A matrix has the Monge property if
for all and . Monge matrices play an important role in
combinatorial optimization; for example, when the transportation problem
(resp., the traveling salesman problem) has a cost matrix which is Monge, then
the problem can be solved in linear (resp., quadratic) time. For given matrix
dimensions, we define the Monge polytope to be the set of nonnegative Monge
matrices normalized with respect to the sum of the entries. In this paper, we
give an explicit description and enumeration of the vertices, edges, and facets
of the Monge polytope; these results are sufficient to construct the face
lattice. In the special case of two-row Monge matrices, we also prove a
polytope volume formula. For symmetric Monge matrices, we show that the Monge
polytope is a simplex and we prove a general formula for its volume