Entropic vector quantile regression: Duality and Gaussian case

TL;DR

This paper explores the theoretical foundations of entropic vector quantile regression (VQR), establishing duality properties, analytic solutions in the Gaussian case, and approximation rates, thereby advancing the understanding of regularized optimal transport problems.

Contribution

It provides the first comprehensive duality theory for entropic VQR, characterizes solutions in the Gaussian case, and quantifies approximation rates to unregularized VQR.

Findings

Strong duality and dual attainment for entropic VQR with unbounded supports.

Dual potentials are real analytic for compactly supported marginals.

Gaussian marginals lead to closed-form Gaussian solutions with known approximation rates.

Abstract

Vector quantile regression (VQR) is an optimal transport (OT) problem subject to a mean-independence constraint that extends classical linear quantile regression to vector response variables. Motivated by computational considerations, prior work has considered entropic relaxation of VQR, but its fundamental structural and approximation properties are still much less understood than entropic OT. The goal of this paper is to address some of these gaps. First, we study duality theory for entropic VQR and establish strong duality and dual attainment for marginals with possibly unbounded supports. In addition, when all marginals are compactly supported, we show that dual potentials are real analytic. Second, building on our duality theory, when all marginals are Gaussian, we show that entropic VQR has a closed-form optimal solution, which is again Gaussian, and establish the precise approximation rate toward unregularized VQR.