Estimating matching affinity matrix under low-rank constraints

TL;DR

This paper introduces a novel method for estimating a low-rank matching affinity matrix in high-dimensional optimal transport problems, revealing key factors influencing matching by using nuclear norm regularization.

Contribution

The paper proposes a new inverse estimation approach with nuclear norm regularization to recover low-rank affinity matrices from observed joint distributions.

Findings

Effective low-rank affinity matrix estimation in high dimensions

Reveals main factors relevant for matching

Improves interpretability of matching models

Abstract

In this paper, we address the problem of estimating transport surplus (a.k.a. matching affinity) in high dimensional optimal transport problems. Classical optimal transport theory specifies the matching affinity and determines the optimal joint distribution. In contrast, we study the inverse problem of estimating matching affinity based on the observation of the joint distribution, using an entropic regularization of the problem. To accommodate high dimensionality of the data, we propose a novel method that incorporates a nuclear norm regularization which effectively enforces a rank constraint on the affinity matrix. The low-rank matrix estimated in this way reveals the main factors which are relevant for matching.