Wasserstein regularization for sparse multi-task regression

Several regression problems encountered in the high-dimensional regime involve the prediction of one (or several) values using a very large number of regressors. In many of these problems, these regressors relate to physical locations, describing for instance measurements taken at neighboring locations, or, more generally quantities that are tied by some underlying geometry: In climate science, regressors may correspond to physical measurements (surface temperature, wind velocity) at different locations across the ocean [Chatterjee et al., 2012]; In genomics, these regressors map to positions on the genome [Laurent et al., 2009]; In functional brain imaging, features correspond to 3D locations in the brain, and a single regression task can correspond to estimating a quantity for a given patient [Owen et al., 2009]. These challenging high-dimensional learning problems have been tackled in recent years using a combination of two approaches: multitask learning to increase the sample size and sparsity. Indeed, it is not uncommon in these problems to aim at predicting several - not just one - related target variables simultaneously. When considering multiple regression tasks, a natural assumption is that prediction functions (and therefore their parameters) for related tasks should share some similarities.