Entropic and displacement interpolation: A computational approach using the Hilbert metric

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 the Schr¨odinger bridge problem (SBP) that leads to entropic interpolation. SBP may be seen as a stochastic regularization of OMT, in fact of a generalization of OMT (allowing for prior dynamics), 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. The actual computation of entropic flows, however, has hardly received any attention. In our recent work on Schroedinger bridges for Markov chains and quantum channels, we showed 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 SBP and ii) provides an efficient computational scheme for both, SBP and OMT. We illustrate this new computational approach by obtaining interpolation of densities in representative examples such as interpolation of images.