Sinkhorn algorithms for entropic vector quantile regression

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.