Schrödinger Bridge Samplers
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Consider a reference Markov process with initial distribution π_0 and transition kernels {M_t}_t∈[1:T], for some T∈N. Assume that you are given distribution π_T, which is not equal to the marginal distribution of the reference process at time T. In this scenario, Schrödinger addressed the problem of identifying the Markov process with initial distribution π_0 and terminal distribution equal to π_T which is the closest to the reference process in terms of Kullback–Leibler divergence. This special case of the so-called Schrödinger bridge problem can be solved using iterative proportional fitting, also known as the Sinkhorn algorithm. We leverage these ideas to develop novel Monte Carlo schemes, termed Schrödinger bridge samplers, to approximate a target distribution π on R^d and to estimate its normalizing constant. This is achieved by iteratively modifying the transition kernels of the reference Markov chain to obtain a process whose marginal distribution at time T becomes closer to π_T = π, via regression-based approximations of the corresponding iterative proportional fitting recursion. We report preliminary experiments and make connections with other problems arising in the optimal transport, optimal control and physics literatures.
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