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Entropic vector quantile regression: Duality and Gaussian case

Kengo Kato, Boyu Wang

TL;DR

Entropic vector quantile regression (VQR) recasts conditional vector quantile estimation as a regularized optimal transport problem with a mean-independence constraint, enabling tractable duality analysis and Gaussian closed-form solutions. The authors prove strong duality and dual attainment under mild moment conditions and establish real analyticity of dual potentials when marginals are compact; in the Gaussian setting they derive a closed-form, Gaussian optimal coupling and quantify convergence to the unregularized VQR with explicit $W_2$-rates. The work extends entropic OT theory to the VQR context, providing rigorous guarantees for unbounded marginals and detailed Gaussian-case insights that inform both theory and computation. Overall, the results enhance the theoretical underpinnings of entropic VQR and offer practical guidance for Gaussian-based modeling and scalable computation.

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.

Entropic vector quantile regression: Duality and Gaussian case

TL;DR

Entropic vector quantile regression (VQR) recasts conditional vector quantile estimation as a regularized optimal transport problem with a mean-independence constraint, enabling tractable duality analysis and Gaussian closed-form solutions. The authors prove strong duality and dual attainment under mild moment conditions and establish real analyticity of dual potentials when marginals are compact; in the Gaussian setting they derive a closed-form, Gaussian optimal coupling and quantify convergence to the unregularized VQR with explicit -rates. The work extends entropic OT theory to the VQR context, providing rigorous guarantees for unbounded marginals and detailed Gaussian-case insights that inform both theory and computation. Overall, the results enhance the theoretical underpinnings of entropic VQR and offer practical guidance for Gaussian-based modeling and scalable computation.

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.
Paper Structure (25 sections, 20 theorems, 147 equations)

This paper contains 25 sections, 20 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Related literature
  4. Organization
  5. Notation
  6. Duality theory for entropic VQR
  7. VQR and entropic VQR
  8. Duality
  9. Regularity of dual potentials
  10. Gaussian case
  11. Closed-form solution
  12. Convergence of entropic VQR
  13. Proofs for Section \ref{['sec: duality']}
  14. Proof of Proposition \ref{['prop: primal attainment']}
  15. Proof of Proposition \ref{['prop: weak duality']}
  16. ...and 10 more sections

Key Result

Proposition 2.1

For every $\varepsilon > 0$, there exists a unique optimal solution $\pi^\varepsilon$ to the primal problem (eq: entropicPrimal).

Theorems & Definitions (34)

  • Proposition 2.1: Primal attainment
  • Proposition 2.2: Weak duality
  • Lemma 2.1
  • Theorem 2.1: Strong duality and dual attainment
  • Proposition 2.3: Uniqueness of dual potentials
  • Proposition 2.4
  • Corollary 2.1
  • Remark 2.1
  • Proposition 2.5
  • Remark 2.2
  • ...and 24 more