  Non-negative matrix and tensor factorisations are a classical tool in machine
learning and data science for finding low-dimensional representations of
high-dimensional datasets. In applications such as imaging, datasets can often
be regarded as distributions in a space with metric structure. In such a
setting, a Wasserstein loss function based on optimal transportation theory is
a natural choice since it incorporates knowledge about the geometry of the
underlying space. We introduce a general mathematical framework for computing
non-negative factorisations of matrices and tensors with respect to an optimal
transport loss, and derive an efficient method for its solution using a convex
dual formulation. We demonstrate the applicability of this approach with
several numerical examples.
