  Learning the multivariate distribution of data is a core challenge in
statistics and machine learning. Traditional methods aim for the probability
density function (PDF) and are limited by the curse of dimensionality. Modern
neural methods are mostly based on black-box models, lacking identifiability
guarantees. In this work, we aim to learn multivariate cumulative distribution
functions (CDFs), as they can handle mixed random variables, allow efficient
box probability evaluation, and have the potential to overcome local sample
scarcity owing to their cumulative nature. We show that any grid sampled
version of a joint CDF of mixed random variables admits a universal
representation as a naive Bayes model via the Canonical Polyadic (tensor-rank)
decomposition. By introducing a low-rank model, either directly in the raw data
domain, or indirectly in a transformed (Copula) domain, the resulting model
affords efficient sampling, closed form inference and uncertainty
quantification, and comes with uniqueness guarantees under relatively mild
conditions. We demonstrate the superior performance of the proposed model in
several synthetic and real datasets and applications including regression,
sampling and data imputation. Interestingly, our experiments with real data
show that it is possible to obtain better density/mass estimates indirectly via
a low-rank CDF model, than a low-rank PDF/PMF model.
