Budget-Constrained Bounds for Mini-Batch Estimation of Optimal Transport (2210.13630v1)

Abstract: Optimal Transport (OT) is a fundamental tool for comparing probability distributions, but its exact computation remains prohibitive for large datasets. In this work, we introduce novel families of upper and lower bounds for the OT problem constructed by aggregating solutions of mini-batch OT problems. The upper bound family contains traditional mini-batch averaging at one extreme and a tight bound found by optimal coupling of mini-batches at the other. In between these extremes, we propose various methods to construct bounds based on a fixed computational budget. Through various experiments, we explore the trade-off between computational budget and bound tightness and show the usefulness of these bounds in computer vision applications.