  The dynamic Schr\"odinger bridge problem provides an appealing setting for
solving constrained time-series data generation tasks posed as optimal
transport problems. It consists of learning non-linear diffusion processes
using efficient iterative solvers. Recent works have demonstrated
state-of-the-art results (eg. in modelling single-cell embryo RNA sequences or
sampling from complex posteriors) but are limited to learning bridges with only
initial and terminal constraints. Our work extends this paradigm by proposing
the Iterative Smoothing Bridge (ISB). We integrate Bayesian filtering and
optimal control into learning the diffusion process, enabling the generation of
constrained stochastic processes governed by sparse observations at
intermediate stages and terminal constraints. We assess the effectiveness of
our method on synthetic and real-world data generation tasks and we show that
the ISB generalises well to high-dimensional data, is computationally
efficient, and provides accurate estimates of the marginals at intermediate and
terminal times.
