Centroid-Referenced Mahalanobis Matching (CRM): A Scalable, Representation-Based Framework for Causal Inference in Large Observational Studies

Abstract

Matching for causal inference can be computationally expensive at scale and can silently change the target population when overlap is limited. We propose Centroid-Referenced Mahalanobis Matching (CRM), which replaces global pairwise search with stratified sampling in two reference coordinates: each unit's Mahalanobis distance from the treated centroid and its Fisher coordinate along the treated-control mean shift. All covariates enter through the treated covariance geometry; CRM is therefore not principal-component preprocessing followed by nearest-neighbor matching. For $n$ units and $p$ pretreatment covariates, its implemented cost is $O(np^2+p^3+n\log n)$, simplifying to $O(np^2+n\log n)$ when $n \ge p$. We derive an error decomposition separating representation, support, discretization, and stochastic components. A pre-matching shortage fraction $\hat{\pi}$ estimates the population support restriction $\pi$, which enters a gap bound under bounded treatment-effect heterogeneity. Final retention is reported separately for capacity-driven exclusions. Under representation sufficiency, smoothness, and adequate cell capacity, CRM has a conservative two-dimensional histogram mean-squared-error bound $O(n_T^{-1/2})$; representation sufficiency is an additional assumption, not a consequence of ignorability given the original covariates. On Criteo, CRM retains at least 99.4% of treated units, has lower MaxSMD than corrected propensity-score matching in 31 of 36 large-scale configurations, and is roughly an order of magnitude faster. Moderate-size simulations favor some pairwise and weighting baselines on balance, locating CRM's contribution in scalability and explicit support diagnostics rather than universal finite-sample dominance.