Statistical Inference for Matching Decisions via Matrix Completion under Dependent Missingness

Abstract: This paper studies decision-making and statistical inference for two-sided matching markets via matrix completion. In contrast to the independent sampling assumed in classical matrix completion literature, the observed entries, which arise from past matching data, are constrained by matching capacity. This matching-induced dependence poses new challenges for both estimation and inference in the matrix completion framework. We propose a non-convex algorithm based on Grassmannian gradient descent and establish near-optimal entrywise convergence rates for three canonical mechanisms, i.e., one-to-one matching, one-to-many matching with one-sided random arrival, and two-sided random arrival. To facilitate valid uncertainty quantification and hypothesis testing on matching decisions, we further develop a general debiasing and projection framework for arbitrary linear forms of the reward matrix, deriving asymptotic normality with finite-sample guarantees under matching-induced dependent sampling. Our empirical experiments demonstrate that the proposed approach provides accurate estimation, valid confidence intervals, and efficient evaluation of matching policies.