Sinkhorn algorithms for entropic vector quantile regression

TL;DR

This paper introduces and analyzes two Sinkhorn algorithms for entropic vector quantile regression, providing convergence guarantees and practical improvements for optimal transport-based statistical modeling.

Contribution

It develops a new Sinkhorn-type algorithm with a projected gradient step for vector quantile regression, along with convergence analysis and explicit bounds.

Findings

Both algorithms converge linearly in dual objective and iterates.

The new projected gradient scheme is more computationally practical.

Explicit bounds on dual potentials and Sinkhorn iterates are derived.

Abstract

Vector quantile regression (VQR) is an optimal transport (OT)-based framework that extends linear quantile regression to vector-valued response variables and can be formulated as an OT problem with a mean-independence constraint. In this paper, we study two Sinkhorn-type algorithms for VQR with entropic regularization, building on our previous work on its duality theory. The first is a direct adaptation of the classical Sinkhorn iteration based on solving the full Schr\"{o}dinger-type system characterizing the dual potentials, which requires solving an implicit functional equation at each iteration. The second algorithm, which is new in the literature, replaces the implicit update with a projected gradient step, resulting in a modified scheme that is computationally more practical. For both algorithms, and for general compactly supported marginals, we establish linear convergence in both the dual objective value and the iterates. A key innovation in our analysis is the derivation of explicit quantitative bounds on the dual potentials and Sinkhorn iterates.