Low rank tensor recovery via iterative hard thresholding

TL;DR

This paper extends iterative hard thresholding algorithms to recover low rank tensors of higher order from limited linear measurements, providing theoretical guarantees and demonstrating effectiveness through numerical experiments.

Contribution

It introduces tensor-specific iterative hard thresholding algorithms for various tensor decompositions and establishes their convergence under a tensor restricted isometry property.

Findings

Subgaussian measurements satisfy tensor RIP with high probability.

Algorithms successfully recover tensors from Gaussian, Fourier, and completion measurements.

Theoretical bounds on the number of measurements needed for accurate recovery.

Abstract

We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank matrix recovery is already well-developed, only few contributions on the low rank tensor recovery problem are available so far. In this paper, we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions, namely the higher order singular value decomposition (HOSVD), the tensor train format (TT), and the general hierarchical Tucker decomposition (HT). We provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank. We show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format. These bounds are extended to partial Fourier maps combined with random sign flips of the tensor entries. Finally, we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from Gaussian random measurements, tensor completion (recovery of missing entries), and Fourier measurements for third order tensors.