Multivariate quantile regression via Kolmogorov-Arnold Networks

This paper introduces a novel algorithm for predicting conditional joint distributions of vector-valued targets in stochastic systems whose randomness is intrinsic rather than arising from observation errors or additive noise. Multivariate quantile regression also involves modeling conditional joint distributions but represents a less challenging task. It predicts the probability that vector-valued targets fall within predefined regions, identifies regions corresponding to predefined probability levels, or performs both tasks simultaneously. The proposed identification technique employs ensembles of Kolmogorov--Arnold networks (KANs) as flexible function approximators. Although the suggested technique is not theoretically restricted to KANs, KANs are particularly well suited to the proposed construction and are therefore used throughout this study. In addition to the training procedure, this work introduces a new discrepancy measure for joint distributions and a goodness-of-fit (GoF) test based on it. This GoF test was initially developed to validate and calibrate the proposed identification technique and is used here in an ad hoc manner. Although the test could be tabulated for broader use, such a tabulation is not pursued in this work. The test is also applicable more generally.