Recovering orthogonal tensors under arbitrarily strong, but locally correlated, noise

TL;DR

This paper introduces a polynomial-time method for recovering orthogonally decomposable tensors corrupted by arbitrarily large, locally correlated noise, even with many missing entries, by solving coupled Sylvester-like equations.

Contribution

It presents a novel tensor completion approach that works under strong, locally correlated noise and high missing data, contrasting with traditional low-rank assumptions.

Findings

Recovery is possible with up to 40% missing entries in high-rank tensors.

The method solves coupled Sylvester-like equations efficiently.

Recovery is feasible even with arbitrarily large noise magnitude.

Abstract

We consider the problem of recovering an orthogonally decomposable tensor with a subset of elements distorted by noise with arbitrarily large magnitude. We focus on the particular case where each mode in the decomposition is corrupted by noise vectors with components that are correlated locally, i.e., with nearby components. We show that this deterministic tensor completion problem has the unusual property that it can be solved in polynomial time if the rank of the tensor is sufficiently large. This is the polar opposite of the low-rank assumptions of typical low-rank tensor and matrix completion settings. We show that our problem can be solved through a system of coupled Sylvester-like equations and show how to accelerate their solution by an alternating solver. This enables recovery even with a substantial number of missing entries, for instance for $n$-dimensional tensors of rank $n$ with up to $40\%$ missing entries.