Iterative Hard Thresholding for Low CP-rank Tensor Models

TL;DR

This paper extends the theoretical guarantees of iterative hard thresholding methods from low Tucker rank tensors to low CP-rank tensors, supported by empirical results demonstrating effectiveness.

Contribution

It introduces a novel extension of tensor recovery guarantees to CP-rank tensors using RIP and recent approximation techniques, advancing tensor recovery theory.

Findings

The method guarantees exact recovery under certain conditions.

Empirical results show promising practical performance.

The approach simplifies assumptions needed for tensor recovery.

Abstract

Recovery of low-rank matrices from a small number of linear measurements is now well-known to be possible under various model assumptions on the measurements. Such results demonstrate robustness and are backed with provable theoretical guarantees. However, extensions to tensor recovery have only recently began to be studied and developed, despite an abundance of practical tensor applications. Recently, a tensor variant of the Iterative Hard Thresholding method was proposed and theoretical results were obtained that guarantee exact recovery of tensors with low Tucker rank. In this paper, we utilize the same tensor version of the Restricted Isometry Property (RIP) to extend these results for tensors with low CANDECOMP/PARAFAC (CP) rank. In doing so, we leverage recent results on efficient approximations of CP decompositions that remove the need for challenging assumptions in prior works. We complement our theoretical findings with empirical results that showcase the potential of the approach.