  Schr\"{o}dinger bridge can be viewed as a continuous-time stochastic control
problem where the goal is to find an optimally controlled diffusion process
whose terminal distribution coincides with a pre-specified target distribution.
We propose to generalize this problem by allowing the terminal distribution to
differ from the target but penalizing the Kullback-Leibler divergence between
the two distributions. We call this new control problem soft-constrained
Schr\"{o}dinger bridge (SSB). The main contribution of this work is a
theoretical derivation of the solution to SSB, which shows that the terminal
distribution of the optimally controlled process is a geometric mixture of the
target and some other distribution. This result is further extended to a time
series setting. One application is the development of robust generative
diffusion models. We propose a score matching-based algorithm for sampling from
geometric mixtures and showcase its use via a numerical example for the MNIST
data set.
