  Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for
geometries in the space of positive densities -- it quantifies the cost of
transporting a mass distribution into another. In particular, it provides
natural options for interpolation of distributions (displacement interpolation)
and for modeling flows. As such it has been the cornerstone of recent
developments in physics, probability theory, image processing, time-series
analysis, and several other fields. In spite of extensive work and theoretical
developments, the computation of OMT for large scale problems has remained a
challenging task. An alternative framework for interpolating distributions,
rooted in statistical mechanics and large deviations, is that of Schroedinger
bridges (entropic interpolation). This may be seen as a stochastic
regularization of OMT and can be cast as the stochastic control problem of
steering the probability density of the state-vector of a dynamical system
between two marginals. In this approach, however, the actual computation of
flows had hardly received any attention. In recent work on Schroedinger bridges
for Markov chains and quantum evolutions, we noted that the solution can be
efficiently obtained from the fixed-point of a map which is contractive in the
Hilbert metric. Thus, the purpose of this paper is to show that a similar
approach can be taken in the context of diffusion processes which i) leads to a
new proof of a classical result on Schroedinger bridges and ii) provides an
efficient computational scheme for both, Schroedinger bridges and OMT. We
illustrate this new computational approach by obtaining interpolation of
densities in representative examples such as interpolation of images.
