Gini-regularized optimal transport with an application to spatio-temporal forecasting

Rapidly growing product lines and services require a finer-granularity forecast that considers geographic locales. However the open question remains, how to assess the quality of a spatio-temporal forecast? In this manuscript we introduce a metric to evaluate spatio-temporal forecasts. This metric is based on an Optimal Transport (OT) problem. The metric we propose is a constrained OT objective function using the Gini impurity function as a regularizer. We demonstrate through computer experiments both the qualitative and the quantitative characteristics of the Gini regularized OT problem. Moreover, we show that the Gini regularized OT problem converges to the classical OT problem, when the Gini regularized problem is considered as a function of λ, the regularization parameter. The convergence to the classical OT solution is faster than the state-of-the-art Entropic-regularized OT[Cuturi, 2013] and results in a numerically more stable algorithm.