AMS Weekly Seminar | Tim Kunisky

Location: Gilman 50

When: September  5th at 1:30 p.m.

Title: Tensor networks, tensor cumulants, and tensor PCA

Abstract: I will discuss a recent line of work whose conceptual goal is to make sense of the questions: What is the right notion of a “spectral algorithm” for tensors? And, how can we understand the power of such algorithms? I will focus on tensor principal component analysis (PCA), a computational problem analogous to, but subtler and more mysterious than, that of identifying and extracting dominant principal components from matrices. I will argue that the right notion of spectral algorithms for tensors, algorithms based on “tensor networks” that compute polynomials of a tensor according to a graphical template, actually form a class much larger and richer than in the matrix case. For some natural models of tensor PCA, I will also show that these spectral algorithms are the best among the even larger class of low-degree polynomial algorithms, which are believed to perform optimally on many statistical tasks.

I will then present some of the tools we have developed to assess the performance of tensor networks on tensor PCA and similar problems. The mathematics behind these tools takes the first steps towards a tantalizing tensor-valued version of free probability theory and especially towards analogs of its notions of free convolution and free cumulants. I will show how a calculus emerges for understanding objects that might be thought of as the “limiting spectra of random tensors,” which allows us to analyze the effect of various deformations of random tensors including as just a first and special case the problem of tensor PCA.

Based on recent joint work with Cris Moore and Alex Wein and ongoing work with Max Jerdee.

Zoom link: https://wse.zoom.us/j/94601022340