Optimal transportation (OT) has had a profound impact on machine learning by providing theoretical and computational tools to realign data sets. In this context, given two large point clouds of sizes north<annotation encoding="application/x-tex”>northnorth and metro<annotation encoding="application/x-tex”>metrometro in Rd<annotation encoding="application/x-tex”>\mathbb{R}^dRdEntropic OT (EOT) solvers have emerged as the most reliable tool to solve the Kantorovich problem and generate a north×metro<annotation encoding="application/x-tex”>n\times mnorth×metro coupling matrix, or to solve Monge's problem and learn a vector-valued forward map. While the robustness of EOT couplings/maps makes them an ideal choice in practical applications, EOT solvers remain difficult to tune due to a small but influential set of hyperparameters, in particular the ubiquitous entropic regularization force. my<annotation encoding="application/x-tex”>\varepsilonmy. Configuration my<annotation encoding="application/x-tex”>\varepsilonmy can be difficult as it simultaneously affects several performance metrics, such as processing speed, statistical performance, generalization, and bias. In this work, we propose a new class of EOT solvers (ProgOT), which can estimate both planes and transport maps. We took advantage of several opportunities to optimize the calculation of EOT solutions by partitioning the mass displacement using a time discretization, drawing inspiration from dynamic formulations of OT (McCann 1997), and conquering each of these steps using EOT with appropriately programmed parameters. We provide experimental evidence demonstrating that ProgOT is a faster and more robust alternative to EOT solvers when computing large-scale couplings and maps, outperforming even neural network-based approaches. We also demonstrate the statistical consistency of ProgOT when estimating OT maps.
