“Statistical Benefits and Computational Challenges of Tensor Spectral Learning”
Given multivariate observations from a statistical model, tensors are a natural way of recording higher order interactions among 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. This approach has recently received a lot of attention for its application to latent variable models. In this talk, I will focus on orthogonally decomposable tensors, which arise naturally in many 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 considerations. 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 based on spectral truncation, which provide rate optimal estimators. Our estimators are also asymptotically normal thus allowing confidence interval construction. Finally I will present some examples demonstrating our theoretical findings.
This talk was held virtually on January 12, 2023 @ 3:00 pm