Plug-in estimation of Schrödinger bridges
This provides a more efficient computational approach for connecting probability distributions in sampling and optimal transport applications.
The authors tackled the problem of estimating Schrödinger bridges between probability distributions without iterative diffusion simulation or neural network training, showing their Sinkhorn bridge method achieves provable convergence rates depending on target measure dimensionality.
We propose a procedure for estimating the Schrödinger bridge between two probability distributions. Unlike existing approaches, our method does not require iteratively simulating forward and backward diffusions or training neural networks to fit unknown drifts. Instead, we show that the potentials obtained from solving the static entropic optimal transport problem between the source and target samples can be modified to yield a natural plug-in estimator of the time-dependent drift that defines the bridge between two measures. Under minimal assumptions, we show that our proposal, which we call the \emph{Sinkhorn bridge}, provably estimates the Schrödinger bridge with a rate of convergence that depends on the intrinsic dimensionality of the target measure. Our approach combines results from the areas of sampling, and theoretical and statistical entropic optimal transport.