Speaker: Arnab Auddy, Ph.D. Candidate in Statistics, Columbia University
Title: Tensor Spectral Learning in Statistics: Benefits and Challenges
Abstract: Given a multivariate observation from a statistical model, tensors are a natural way of recording higher order interactions among the variables. Tensor spectral learning is a collection of methods wherein we aim to decompose a tensor into its components, each of which correspond to interpretable features of the model. In this talk, I will focus on orthogonally decomposable tensors, which arise naturally in many statistical problems. These tensors have a decomposition that can be interpreted very similarly to matrix SVD, but automatically provides much better identifiability properties than their matrix counterparts. I will show that in such a tensor decomposition, a small perturbation affects each singular vector in isolation, and their estimatibility does not depend on the gap between consecutive singular values. In contrast to these attractive statistical properties, in general, tensor methods present us with intriguing computational challenges. I will illustrate these phenomena in the particular application to a spiked tensor PCA problem and in Independent Component Analysis (ICA). Interestingly there is a gap within the information theoretic and computationally tractable limits of both problems. Above the computational threshold, we provide noise robust algorithms and obtain rate optimal estimators. Our estimators are also asymptotically normal thus allowing confidence interval construction. Finally I will present some examples demonstrating our theoretical findings.
Note: Seminars are free and open to the public. Reception to follow.
