Conditional Generative Quantile Networks via Optimal Transport and Convex Potentials

Quantile regression has a natural extension to generative modelling by leveraging a stronger convergence in pointwise rather than in distribution. While the pinball quantile loss works in the scalar case, it does not have a provable extension to the vector case. In this work, we consider a quantile approach to generative modelling using optimal transport with provable guarantees. We suggest and prove that by optimizing smooth functions with respect to the dual of the correlation maximization problem, the optimum is convex almost surely and hence construct a Brenier map as our generative quantile network. Furthermore, we introduce conditional generative modelling with a Kantorovich dual objective by constructing an affine latent model with respect to the covariates. Through extensive experiments on synthetic and real datasets for conditional generative and probabilistic forecasting tasks, we demonstrate the efficacy and versatility of our theoretically motivated model as a distribution estimator and conditioner.