Estimating Higher-Order Mixed Memberships via the ℓ_2,∞ Tensor Perturbation Bound

by   Joshua Agterberg, et al.

Higher-order multiway data is ubiquitous in machine learning and statistics and often exhibits community-like structures, where each component (node) along each different mode has a community membership associated with it. In this paper 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 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 on estimation accuracy. To prove our consistency result, we develop the ℓ_2,∞ 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. Whereas other leave-one-out analyses typically focus on sequences constructed by analyzing the output of a given algorithm with a small part of the noise removed, our leave-one-out analysis constructions use both the previous iterates and the additional tensor structure to eliminate a potential additional source of error. Finally, we apply our methodology to real and simulated data, including applications to two flight datasets and a trade network dataset, demonstrating some effects not identifiable from the model with discrete community memberships.


page 16

page 17

page 18

page 20


A Sharp Blockwise Tensor Perturbation Bound for Orthogonal Iteration

In this paper, we develop novel perturbation bounds for the high-order o...

Tensor SVD: Statistical and Computational Limits

In this paper, we propose a general framework for tensor singular value ...

The ℓ_∞ Perturbation of HOSVD and Low Rank Tensor Denoising

The higher order singular value decomposition (HOSVD) of tensors is a ge...

Optimal High-order Tensor SVD via Tensor-Train Orthogonal Iteration

This paper studies a general framework for high-order tensor SVD. We pro...

Optimizing Orthogonalized Tensor Deflation via Random Tensor Theory

This paper tackles the problem of recovering a low-rank signal tensor wi...

Reduced Higher Order SVD: ubiquitous rank-reduction method in tensor-based scientific computing

Tensor numerical methods, based on the rank-structured tensor representa...

On the Theoretical Properties of the Network Jackknife

We study the properties of a leave-node-out jackknife procedure for netw...

Please sign up or login with your details

Forgot password? Click here to reset