Tensor principal component pursuit (TPCP) is a powerful approach in the tensor robust principal component analysis (TRPCA), where the goal is to decompose a data tensor to a low-tubal-rank part plus a sparse residual. TPCP is shown to be effective under certain tensor incoherence conditions, which can be restrictive in practice. In this paper, we propose a Modified-TPCP, which incorporates the prior subspace information in the analysis. With the aid of prior info, the proposed method is able to recover the low-tubal-rank and the sparse components under a significantly weaker incoherence assumption. We further design an efficient algorithm to implement Modified-TPCP based upon the alternating direction method of multipliers (ADMM). The promising performance of the proposed method is supported by simulations and real data applications.

This letter considers the recovery of a low-tubal-rank tensor from incomplete linear observations. It is shown that the unknown tensor Z ∈ Rn1×n2×n3of tubal-rank r can be reconstructed as a unique solution of a tractable method - tensor nuclear norm (TNN) minimization, provided that the number of Gaussian observations m ≥ 3r(n1+ n2- r)n3+ 1. In this work, we examine the fundamental question of the minimal number of linear observations needed to reconstruct the tensor Z from these observations, regardless of the practicality of the reconstruction scheme. Consequently, we provide two benchmark results so that different reconstruction schemes including TNN minimization can be compared to each other. Specifically, we conclude that m ≥ 2r(n1+ n2- 2r)n3and m ≥ r(n1+ n2- r)n3+ 1 Gaussian observations are necessary and sufficient to guarantee uniform recovery and nonuniform recovery using tensor tubalrank minimization method, respectively.

This work gains a sharp sufficient condition on the block restricted isometry property for the recovery of sparse signal and corresponding upper bound estimate of error. Under the certain assumption, the signal with block structure can be stably recovered in the presence of noisy case and the block sparse signal can be exactly reconstructed in the noise‐free case. Besides, an example is proposed to exhibit the condition is sharp. Numerical simulations are carried out to demonstrate that authors’ results are verifiable and l 2 / l 1 minimisation method is robust and stable for the recovery of block sparse signals.

Motivated by the recently emerged method for sparse signal recovery, in this study, the authors make an ongoing effect to extend this methodology to the setting of block sparsity, which directly leads to the proposed method for block‐sparse signal recovery. Some theoretical results are induced to guarantee the validity of proposed method. In particular, the obtained recovery condition rigorously includes the one induced by Yin et al ., and the obtained error estimate can be used to model both the (block‐) sparse and non‐sparse signals, which is more comprehensive than that induced by Yin et al . which applies only to the sparse signals. The authors also derive an alternating direction method of multipliers (ADMM)‐based algorithm to tackle the induced optimisation problem. Some experimental results that are based on the synthetic block‐sparse signals and the real‐world foetal electrocardiogram signals further demonstrate the better performance of the method when it is compared with the state‐of‐the‐art group‐lasso method and method for 0 < q < 1.

Compressed sensing is a novel theory for signal sampling, which breaks through Nyquist/Shannon sampling limitation and makes it into reality that one can efficiently collect and robustly reconstruct a sparse signal. However, some signals exhibit additional structures in some redundant dictionaries, which is called block‐sparse signal. In this study, non‐convex block‐sparse compressed sensing with redundant dictionaries is investigated. Under the block D‐RIP condition , a sufficient condition for robust signal reconstruction with redundant dictionaries by mixed minimisation is established. Furthermore, the authors’ theoretical results show that, under the assumption that , , where urn:x-wiley:17519675:media:sil2bf00440:sil2bf00440-math-0005 then the block k ‐sparse signal can be stably reconstructed via non‐convex ℓ 2 /ℓ p minimisation with redundant dictionaries in the presence of noise. Particularly, this improves the existed result when the block‐sparse signal degenerate to the conventional signal case. Besides, the authors also obtain robust reconstruction condition and error upper bound estimation when the block number is no more than four times the sparsity of the block signal . Moreover, the numerical experiments to some extent testify the performance of non‐convex minimisation with redundant dictionaries.

By embedding an ℓp-norm noise constraint for p ≥ 2 into the recently emerged ℓ1-2method, in this letter, we study theoretically and numerically an ℓ1-2/ℓpmethod for recovery of general noisy signals from highly coherent measurement matrices. In particular, the obtained theoretical results not only improve the condition deduced in [1] for Gaussian noisy signal recovery but also provide a new theoretical guarantee for generally nonGaussian noisy signal recovery. What is more, to better boost the recovery performance, a partial sum ℓ1-2/ℓpmethod is also proposed latter. This improved method, together with the previous ℓ1-2/ℓpmethod, becomes more competitive when compared with some of the state-of-the-art methods in recovering noisy signals from highly coherent measurement matrices.