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In this paper, we describe a novel iterative procedure called SISTA to learnthe underlying cost in optimal transport problems. SISTA is a hybrid betweentwo classical methods, coordinate descent ("S"-inkhorn) and proximal gradientdescent ("ISTA"). It alternates between a phase of exact minimization over thetransport potentials and a phase of proximal gradient descent over theparameters of the transport cost. We prove that this method converges linearly,and we illustrate on simulated examples that it is significantly faster thanboth coordinate descent and ISTA. We apply it to estimating a model ofmigration, which predicts the flow of migrants using country-specificcharacteristics and pairwise measures of dissimilarity between countries. Thisapplication demonstrates the effectiveness of machine learning in quantitativesocial sciences.

SISTA: learning optimal transport costs under sparsity constraints