  We focus in this paper on high-dimensional regression problems where each
regressor can be associated to a location in a physical space, or more
generally a generic geometric space. Such problems often employ sparse priors,
which promote models using a small subset of regressors. To increase
statistical power, the so-called multi-task techniques were proposed, which
consist in the simultaneous estimation of several related models. Combined with
sparsity assumptions, it lead to models enforcing the active regressors to be
shared across models, thanks to, for instance L1 / Lq norms. We argue in this
paper that these techniques fail to leverage the spatial information associated
to regressors. Indeed, while sparse priors enforce that only a small subset of
variables is used, the assumption that these regressors overlap across all
tasks is overly simplistic given the spatial variability observed in real data.
In this paper, we propose a convex regularizer for multi-task regression that
encodes a more flexible geometry. Our regularizer is based on unbalanced
optimal transport (OT) theory, and can take into account a prior geometric
knowledge on the regressor variables, without necessarily requiring overlapping
supports. We derive an efficient algorithm based on a regularized formulation
of OT, which iterates through applications of Sinkhorn's algorithm along with
coordinate descent iterations. The performance of our model is demonstrated on
regular grids with both synthetic and real datasets as well as complex
triangulated geometries of the cortex with an application in neuroimaging.

Machine Learning;Machine Learning