Normalized Iterative Hard Thresholding for Tensor Recovery

Abstract: Low-rank recovery builds upon ideas from the theory of compressive sensing, which predicts that sparse signals can be accurately reconstructed from incomplete measurements. Iterative thresholding-type algorithms-particularly the normalized iterative hard thresholding (NIHT) method-have been widely used in compressed sensing (CS) and applied to matrix recovery tasks. In this paper, we propose a tensor extension of NIHT, referred to as TNIHT, for the recovery of low-rank tensors under two widely used tensor decomposition models. This extension enables the effective reconstruction of high-order low-rank tensors from a limited number of linear measurements by leveraging the inherent low-dimensional structure of multi-way data. Specifically, we consider both the CANDECOMP/PARAFAC (CP) rank and the Tucker rank to characterize tensor low-rankness within the TNIHT framework. At the same time, we establish a convergence theorem for the proposed TNIHT method under the tensor restricted isometry property (TRIP), providing theoretical support for its recovery guarantees. Finally, we evaluate the performance of TNIHT through numerical experiments on synthetic, image, and video data, and compare it with several state-of-the-art algorithms.