Stochastic control, entropic interpolation and gradient flows on
Wasserstein product spaces
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by
Yongxin Chen, Tryphon Georgiou, Michele Pavon
2016
Abstract
Since the early nineties, it has been observed that the Schroedinger bridge
problem can be formulated as a stochastic control problem with atypical
boundary constraints. This in turn has a fluid dynamic counterpart where the
flow of probability densities represents an entropic interpolation between the
given initial and final marginals. In the zero noise limit, such entropic
interpolation converges in a suitable sense to the displacement interpolation
of optimal mass transport (OMT). We consider two absolutely continuous curves
in Wasserstein space W_2 and study the evolution of the relative
entropy on W_2× W_2 on a finite time interval. Thus, this
study differs from previous work in OMT theory concerning relative entropy from
a fixed (often equilibrium) distribution (density). We derive a gradient flow
on Wasserstein product space. We find the remarkable property that fluxes in
the two components are opposite. Plugging in the "steepest descent" into the
evolution of the relative entropy we get what appears to be a new formula: The
two flows approach each other at a faster rate than that of two solutions of
the same Fokker-Planck. We then study the evolution of relative entropy in the
case of uncontrolled-controlled diffusions. In two special cases of the
Schroedinger bridge problem, we show that such relative entropy may be
monotonically decreasing or monotonically increasing.
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