Quadratically regularized OT

The quadratically regularized optimal transport problem (QOT) has emerged in the literature as a sparse alternative to entropic regularization (EOT). Unlike EOT, whose solutions always have full support—even for small regularization parameters—QOT solutions, or QOT plans, tend to approximate the support of the unregularized transport problem. This raises natural questions: Do the supports decrease monotonically? At what rate does this support reduction occur? How quickly does the QOT cost converge to the optimal transport cost? In this talk, we will review recent theoretical results addressing these questions.