Calibrated Multiple-Output Quantile Regression with Representation Learning

We develop a method to generate predictive regions that cover a multivariate\nresponse variable with a user-specified probability. Our work is composed of\ntwo components. First, we use a deep generative model to learn a representation\nof the response that has a unimodal distribution. Existing multiple-output\nquantile regression approaches are effective in such cases, so we apply them on\nthe learned representation, and then transform the solution to the original\nspace of the response. This process results in a flexible and informative\nregion that can have an arbitrary shape, a property that existing methods lack.\nSecond, we propose an extension of conformal prediction to the multivariate\nresponse setting that modifies any method to return sets with a pre-specified\ncoverage level. The desired coverage is theoretically guaranteed in the\nfinite-sample case for any distribution. Experiments conducted on both real and\nsynthetic data show that our method constructs regions that are significantly\nsmaller compared to existing techniques.\n