COMPUTING ALL OPTIMAL PARTIAL TRANSPORTS

Abstract

We consider the classical version of the optimal partial transport problem. Let µ (with a mass of U ) and ν (with a mass of S) be two discrete mass distributions with S ≤ U and let n be the total number of points in the supports of µ and ν. For a parameter α ∈ [0, S], consider the minimum-cost transport plan σ α that transports a mass of α from ν to µ. An OT-profile captures the behavior of the cost of σ α as α varies from 0 to S. There is only limited work on OT-profile and its mathematical properties (see Figalli ( 2010)). In this paper, we present a novel framework to analyze the properties of the OT-profile and also present an algorithm to compute it. When µ and ν are discrete mass distributions, we show that the OT-profile is a piecewise-linear non-decreasing convex function. Let K be the combinatorial complexity of this function, i.e., the number of line segments required to represent the OT-profile. Our exact algorithm computes the OT-profile in Õ(n 2 K) time. Given δ > 0, we also show that the algorithm by Lahn et al. ( 2019) can be used to δ-approximate the OT-profile in O(n 2 /δ + n/δ 2 ) time. This approximation is a piecewise-linear function of a combinatorial complexity of O(1/δ). An OT-profile is arguably more valuable than the OT-cost itself and can be used within applications. Under a reasonable assumption of outliers, we also show that the first derivative of the OT-profile sees a noticeable rise before any of the mass from outliers is transported. By using this property, we get an improved prediction accuracy for an outlier detection experiment. We also use this property to predict labels and estimate the class priors within PU-Learning experiments. Both these experiments are conducted on real datasets.

1. INTRODUCTION

Given two discrete probability distributions µ (with a mass of U = 1) with the set A as the support and ν (with a mass of S = 1) with B as the support, where |A| + |B| = n, in the optimal transport problem, one wishes to compute the minimum cost plan to transport mass from ν to µ. When the mass U ̸ = S, the problem is called the unbalanced optimal transport. In the partial optimal transport problem, given a parameter α ∈ [0, S], one wishes to determine the α-optimal partial transport cost which is the minimum work required to transport a mass of α from ν to µ. Owing to its strong statistical properties, the optimal transport cost (Villani ( 2003 The exact optimal transport cost and plan, including in the unbalanced case, can be computed in O(n 3 log n) time. For a fixed value of α ∈ [0, S], one can easily reduce the problem of computing an * Following convention from Theoretical Computer Science, all authors are ordered in alphabetical order 1



); Peyré & Cuturi (2019)) is considered to be an attractive dissimilarity metric between probability distributions, and has found numerous applications in areas involving GANs, image processing, (Arjovsky et al. (2017); Liu et al. (2018); Balaji et al. (2020); Lin et al. (2021); Schmitz et al. (2018); Chen et al. (2019)), variational inference (Ambrogioni et al. (2018)), econometrics (Galichon (2016)) and other areas of natural science (Schiebinger et al. (2019); Sun et al. (2020)) and applied mathematics (Santambrogio (2015)). Similarly the unbalanced and partial optimal transport has been used for various problems that arise in machine learning, including GAN training, image processing, outlier detection and Positive Unlabelled (PU-) learning (Yang & Uhler (2018); Bonneel & Coeurjolly (2019); Chapel et al. (2020); Mukherjee et al. (2021)).

