  Optimal transportation distances are a fundamental family of parameterized
distances for histograms. Despite their appealing theoretical properties,
excellent performance in retrieval tasks and intuitive formulation, their
computation involves the resolution of a linear program whose cost is
prohibitive whenever the histograms' dimension exceeds a few hundreds. We
propose in this work a new family of optimal transportation distances that look
at transportation problems from a maximum-entropy perspective. We smooth the
classical optimal transportation problem with an entropic regularization term,
and show that the resulting optimum is also a distance which can be computed
through Sinkhorn-Knopp's matrix scaling algorithm at a speed that is several
orders of magnitude faster than that of transportation solvers. We also report
improved performance over classical optimal transportation distances on the
MNIST benchmark problem.
