Abstract
Vector quantile regression (VQR) estimates the conditional vector quantile function (CVQF), a fundamental quantity which fully represents the conditional distribution of Y|X. VQR is formulated as an optimal transport (OT) problem between a uniform r.v. U and the target r.v. Y|X , the solution of which is a unique transport map, co-monotonic with U. Recently NL-VQR has been proposed to estimate support non-linear CVQFs, together with fast solvers which enabled the use of this tool in practical applications. Despite its utility, the scalability and estimation quality of NL-VQR is limited due to a discretization of the OT problem onto a grid of quantile levels. We propose a novel continuous formulation and parametrization of VQR using partial input-convex neural networks (PICNNs). Our approach allows for accurate, scalable, differentiable and invertible estimation of non-linear CVQFs. We further demonstrate, theoretically and experimentally, how continuous CVQFs can be used for general statistical inference tasks such as estimation of likelihoods, CDFs, confidence sets, coverage, sampling, and more. This work is an important step towards unlocking the full potential of VQR.
Irene Tallini
PhD Student
I’m a Computer Science PhD with Math Bachelor and passion. Right now I’m working on AI for music and vector quantile regression. I like to sing, also.
Marco Pegoraro
PhD Student
I am a Ph.D. student in Geometric Deep Learning. My research activity is focused on spectral geometry processing applied to graph learning and computational biology.
