Low rank tensor recovery via iterative hard thresholding

• Published in: Linear algebra and its applications Vol. 523; pp. 220 - 262
• Main Authors: Rauhut, Holger, Schneider, Reinhold, Stojanac, Željka
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15-06-2017; American Elsevier Company, Inc
• Subjects: Gaussian random ensemble; Hierarchical tensor format; Higher order singular value decomposition; Iterative hard thresholding; Linear algebra; Low rank recovery; Random partial Fourier ensemble; Tensor completion; Tensor decompositions; Tensor train decomposition; Higher order singular value decomposition; Iterative hard thresholding; Random partial Fourier ensemble; Hierarchical tensor format; 15A69; Low rank recovery; Tensor train decomposition; 15B52; Tensor completion; Gaussian random ensemble; 65Fxx; Tensor decompositions; 94A20
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2017.02.028

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.