Seminar by Joshua Agterberg

Please Note: This seminar will be given in person.

Estimating Higher-Order Mixed Memberships via the Two to Infinity Tensor Perturbation Bound

Higher-order multiway data is ubiquitous in machine learning and statistics, and there is a need to develop new methodologies for these types of data that succinctly capture the underlying structures.  For example, multiway data may exhibit community-like structure, where each component (node) along each different mode has a community membership associated to it. We propose the tensor mixed-membership blockmodel, a generalization of the tensor blockmodel positing that memberships need not be discrete, but instead are convex combinations of latent communities. We establish the identifiability of our model, and we propose a computationally efficient estimation procedure based on the higher-order orthogonal iteration algorithm (HOOI) for tensor SVD composed with a simplex corner-finding algorithm.  We then demonstrate the consistency of our estimation procedure by providing a per-node error bound, which showcases the effect of higher-order structures in estimation accuracy. To prove our consistency result, we develop the $\ell_{2,\infty}$ tensor perturbation bound for HOOI under independent, possibly heteroskedastic, subgaussian noise that may be of independent interest. Our analysis uses a novel leave-one-out construction for the iterates, and our bounds depend only on spectral properties of the underlying low-rank tensor under nearly optimal signal-to-noise ratio conditions such that tensor SVD is computationally feasible. Finally, we apply our methodology to US flight data, showcasing the effect of COVID-19 on flights.