  For a very broad range of problems, recommendation algorithms have been
increasingly used over the past decade. In most of these algorithms, the
predictions are built upon user-item affinity scores which are obtained from
high-dimensional embeddings of items and users. In more complex scenarios, with
geometrical or capacity constraints, prediction based on embeddings may not be
sufficient and some additional features should be considered in the design of
the algorithm. In this work, we study the recommendation problem in the setting
where affinities between users and items are based both on their embeddings in
a latent space and on their geographical distance in their underlying euclidean
space (e.g., $\mathbb{R}^2$), together with item capacity constraints. This
framework is motivated by some real-world applications, for instance in
healthcare: the task is to recommend hospitals to patients based on their
location, pathology, and hospital capacities. In these applications, there is
somewhat of an asymmetry between users and items: items are viewed as static
points, their embeddings, capacities and locations constraining the allocation.
Upon the observation of an optimal allocation, user embeddings, items
capacities, and their positions in their underlying euclidean space, our aim is
to recover item embeddings in the latent space; doing so, we are then able to
use this estimate e.g. in order to predict future allocations. We propose an
algorithm (SiMCa) based on matrix factorization enhanced with optimal transport
steps to model user-item affinities and learn item embeddings from observed
data. We then illustrate and discuss the results of such an approach for
hospital recommendation on synthetic data.
