  During recent decades, there has been a substantial development in optimal
mass transport theory and methods. In this work, we consider multi-marginal
problems wherein only partial information of each marginal is available, which
is a setup common in many inverse problems in, e.g., imaging and spectral
estimation. By considering an entropy regularized approximation of the original
transport problem, we propose an algorithm corresponding to a block-coordinate
ascent of the dual problem, where Newton's algorithm is used to solve the
sub-problems. In order to make this computationally tractable for large-scale
settings, we utilize the tensor structure that arises in practical problems,
allowing for computing projections of the multi-marginal transport plan using
only matrix-vector operations of relatively small matrices. As illustrating
examples, we apply the resulting method to tracking and barycenter problems in
spatial spectral estimation. In particular, we show that the optimal mass
transport framework allows for fusing information from different time steps, as
well as from different sensor arrays, also when the sensor arrays are not
jointly calibrated. Furthermore, we show that by incorporating knowledge of
underlying dynamics in tracking scenarios, one may arrive at accurate spectral
estimates, as well as faithful reconstructions of spectra corresponding to
unobserved time points.
