Optimal Transport, a theory for optimal allocation of resources, is widely used in various fields such as astrophysics, machine learning, and imaging science. However, many applications impose elementwise constraints on the transport plan which traditional optimal transport cannot enforce. Here we introduce Supervised Optimal Transport (sOT) that formulates a constrained optimal transport problem where couplings between certain elements are prohibited according to specific applications.  sOT is proved to be equivalent to an $l^1$ penalized optimization problem, from which efficient algorithms are designed to solve its entropy regularized formulation. We demonstrate the capability of sOT by comparing it to other variants and extensions of traditional OT in color transfer problem. We also study the barycenter problem in sOT formulation, where we discover and prove a unique reverse and portion selection (control) mechanism. Supervised optimal transport is broadly applicable to applications in which constrained transport plan is involved and the original unit should be preserved by avoiding normalization.

赵雁翔老师于2011年博士毕业于宾州州立大学, 2011-2014年于加州大学圣地亚哥分校从事博士后工作。2014年至今任职于乔治华盛顿大学数学系。现担任该校数学系主管研究生工作的副系主任。赵雁翔老师的科研方向主要是科学计算, 数学建模在生物, 物理, 材料学等学科领域的应用。