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Tensors and Probability: an Intriguing Union

Nikos Sidiropoulos, Louis T. Rader Professor and Chair of the Electrical and Computer Engineering Department at the University of Virginia

Wegmans Hall 1400

Abstract: We reveal an interesting link between tensors and multivariate statistics. The rank of a multivariate probability tensor can be interpreted as a nonlinear measure of statistical dependence of the associated random variables. Rank equals one when the random variables are independent, and complete statistical dependence corresponds to full rank. In practice we usually work with random variables that are neither independent nor fully dependent—partial dependence is typical, and can be modeled using a low-rank multivariate probability tensor. Directly estimating such a tensor from sample averages is impossible even for as few as ten random variables taking ten values each—yielding ten billion unknowns; but we often have enough data to estimate lower-order marginalized distributions. We prove that it is possible to identify the higher-order joint probabilities from lower order ones, provided that the higher-order probability tensor has low-enough rank, i.e., the random variables are only partially dependent. We also provide a computational identification algorithm that is shown to work well on both simulated and real data. The insights and results have numerous applications in estimation, hypothesis testing, completion, machine learning, and system identification. Low-rank tensor modeling thus provides a `universal' non-parametric (model-free) alternative to probabilistic graphical models.

Bio: Nikos Sidiropoulos is the Louis T. Rader Professor and Chair of the Electrical and Computer Engineering Department at the University of Virginia. He earned his Ph.D. in Electrical Engineering from the University of Maryland–College Park, in 1992. He has served on the faculty of the University of Virginia, University of Minnesota, and the Technical University of Crete, Greece. His research interests are in signal processing, optimization, and tensor decomposition, with applications in machine learning and communications. He received the NSF/CAREER award in 1998, the IEEE Signal Processing Society (SPS) Best Paper Award in 2001, 2007, and 2011, served as IEEE SPS Distinguished Lecturer (2008-2009), and currently serves as Vice President - Membership of IEEE SPS. He received the 2010 IEEE Signal Processing Society Meritorious Service Award, and the 2013 Distinguished Alumni Award from the University of Maryland, Dept. of ECE. He is a Fellow of IEEE (2009) and a Fellow of EURASIP (2014).

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