Tensors, Learning, and “Kolmogorov Extension” for Finite-Alphabet Random Vectors

Estimating the joint probability mass function (PMF) of a set of random variables lies at the heart of statistical learning and signal processing. Without structural assumptions, such as modeling the variables as a Markov chain, tree, or other graphical model, joint PMF estimation is often considered mission impossible-the number of unknowns grows exponentially with the number of variables. But who gives us the structural model? What if we only observe random subsets of the variables, can we still reliably estimate the joint PMF of all? This paper shows, perhaps surprisingly, that if the joint PMF of any three variables can be estimated, then the joint PMF of all the variables can be provably recovered under relatively mild conditions. The result is reminiscent of Kolmogorov's extension theorem - consistent specification of lower-dimensional distributions induces a unique probability measure for the entire process. The difference is that for processes of limited complexity (rank of the high-dimensional PMF) it is possible to obtain complete characterization from only three-dimensional distributions. In fact not all three-dimensional PMFs are needed; and under more stringent conditions even two-dimensional will do. Exploiting multilinear (tensor) algebra, this paper proves that such higher-dimensional PMF completion can be guaranteed-several pertinent identifiability results are derived. It also provides a practical and efficient algorithm to carry out the recovery task. Judiciously designed simulations and real-data experiments on movie recommendation and data classification are presented to showcase the effectiveness of the approach.