In recent years, tensor decomposition-based approaches forhyperspectral anomaly detection(HAD) have gained significant attention in the field of remote sensing. However, existing methods often fail to flexibly and effectively extract both the global correlations and local smoothness of the background components inhyperspectral images(HSIs). To mitigate this critical issue, we put forward a novel HAD method named HAD-EUNTRFR, which incorporates an enhanced unified nonconvex tensor ring (TR) factors regularization. In the HAD-EUNTRFR framework, the raw HSIs are first decomposed into background and anomaly components using the idea of tensor robust principal component analysis. The TR decomposition is then employed to capture the spatial-spectral correlations within the background component. Additionally, we introduce a unified and efficient nonconvex regularizer, induced bytensor singular value decomposition(T-SVD), to simultaneously encode the low-rankness and sparsity of the 3-D gradient TR factors into a unique concise form. The above characterization scheme enables the interpretable gradient TR factors to inherit the low-rankness and smoothness of the original background. To further enhance anomaly detection, we design a generalized nonconvex regularization term to exploit the group sparsity of the anomaly component. Based upon the above, we ultimately propose a scalable and reliable nonconvex HAD model. To solve the resulting doubly nonconvex model, we develop a highly efficient optimization algorithm based on thealternating direction method of multipliers(ADMM) framework. Theoretical results on convergence analysis for the proposed algorithm are derived. Experimental results on several benchmark datasets demonstrate that our proposed method outperforms existingstate-of-the-art(SOTA) approaches in terms of detection accuracy.

Anomaly detection in hyperspectral images (HSIs) aims to identify sparse, interesting anomalies against the background, which has become a significant topic in remote sensing. Although the existing tensor-based methods have achieved commendable performance to some extent, there is still room for further improvement. In combination with three key techniques, i.e., gradient map-based modeling, circular tensor ring (TR) unfolding, and nonconvex regularization, this article proposes a novel generalized nonconvex method for hyperspectral anomaly detection (HAD) tasks within the TR framework. For the implementation of our proposed approach, abbreviated as TR-GNHAD, we first develop an effective and reliable HAD model in virtue of two newly unified nonconvex regularizers. The first regularizer is devised under a new prior characterization paradigm, which has a strong ability to encode two insightful prior information underlying the HSI’s background simultaneously, i.e., global low rankness and local smoothness. The other regularizer can well capture the structured sparsity of the abnormal component. Then, we derive an efficient optimization algorithm to solve the proposed model based on the alternating direction method of multipliers (ADMMs) framework. Experiments conducted on 12 HSI datasets illustrate that the proposed approach achieves highly competitive performance in both qualitative and quantitative metrics compared with several state-of-the-art HAD methods.

Within the tensor singular value decomposition (T-SVD) framework, existing robust low-rank tensor completion approaches have made great achievements in various areas of science and engineering. Nevertheless, these methods involve the T-SVD based low-rank approximation, which suffers from high computational costs when dealing with large-scale tensor data. Moreover, most of them are only applicable to third-order tensors. Against these issues, in this article, two efficient low-rank tensor approximation approaches fusing random projection techniques are first devised under the order-d ( d ≥ 3 ) T-SVD framework. Theoretical results on error bounds for the proposed randomized algorithms are provided. On this basis, we then further investigate the robust high-order tensor completion problem, in which a double nonconvex model along with its corresponding fast optimization algorithms with convergence guarantees are developed. Experimental results on large-scale synthetic and real tensor data illustrate that the proposed method outperforms other state-of-the-art approaches in terms of both computational efficiency and estimated precision.

In the real world, a large amount of high-order tensor data (order>3) exists, such as color videos, multispectral videos, and light-field images. However, these data often face challenges in transportation, storage, and susceptibility to damage. Meanwhile, most existing tensor-based information processing methods only concentrate on third-order tensors, which may not meet the complex requirements of high-dimensional data processing. In this paper, to better address the high-order tensor recovery issue, we propose a novel method that couples multilayer subspace priors with high-order tensor recovery techniques for tensor completion and robust tensor principal component analysis. Moreover, we provide theoretical guarantees for our approach's recovery and demonstrate that it achieves comparable performance under weaker incoherent conditions. Additionally, we develop two efficient and interpretable algorithms based on the alternating direction method of multipliers (ADMM) to solve our model. Owing to the adaptability of subspace prior information, our method demonstrates superior performance in recovering various types of data, including color videos and multispectral videos, compared with various advanced algorithms currently available.

Tensor recovery is a fundamental problem in tensor research field. It generally requires to explore intrinsic prior structures underlying tensor data, and formulate them as certain forms of regularization terms for guiding a sound estimate of the restored tensor. Recent researches have made significant progress by adopting two insightful tensor priors, i.e., global low-rankness (L) and local smoothness (S), which are always encoded as a sum of two separate regularizers into recovery models. However, unlike the primary theoretical developments on low-rank tensor recovery, these joint "L+S" models have no theoretical exact-recovery guarantees yet, making the methods lack reliability in real practice. To this crucial issue, in this work, we build a unique regularizer termed as tensor correlated total variation (t-CTV), which essentially encodes both L and S priors of a tensor simultaneously. Especially, by equipping t-CTV into the recovery models, we can rigorously prove the exact recovery guarantees for two typical tensor recovery tasks, i.e., tensor completion and tensor robust principal component analysis. To the best of our knowledge, this should be the first exact-recovery results among all related "L+S" methods for tensor recovery. We further propose ADMM algorithms with fine convergence to solve the proposed models. Significant recovery accuracy improvements are observed in extensive experiments. Typically, our method achieves a workable performance when the missing rate is extremely large, e.g., 99.5%, for the color image inpainting task, while all its peers totally fail in such a challenging case. Code is released at https://github.com/wanghailin97.

Currently, low-rank tensor recovery employing the subspace prior information is an emerging topic, which has attracted considerable attention. However, existing studies cannot flexibly and fully utilize the accessible subspace prior information, thereby leading to suboptimal restored performance. Aiming at addressing this issue, based on the tensor singular value decomposition (t-SVD), this article presents a novel strategy that integrates more than two layers of subspace knowledge about columns and rows of target tensor into one unified recovery framework. Specially, we first design a multilayer subspace prior learning scheme, and then apply it to two common low-rank tensor recovery problems, i.e., tensor completion and tensor robust component principal analysis. Crucially, we prove that our approach can achieve exact recovery of tensors under a significantly weaker incoherence assumption than the analogous conditions previously proposed. Furthermore, two efficient algorithms with convergence guarantees based on alternating direction method of multipliers (ADMM) are proposed to solve the corresponding models. The experimental results on synthetic and real tensor data show that the proposed algorithms outperform other state-of-the-art algorithms in terms of both qualitative and quantitative metrics.

The latest tensor recovery methods based on tensor Singular Value Decomposition (t-SVD) mainly utilize the tensor nuclear norm (TNN) as a convex surrogate of the rank function. However, TNN minimization treats each rank component equally and tends to over-shrink the dominant ones, thereby usually leading to biased solutions. To handle this critical issue, we put forward a weighted tensor Schantten-p (0q(0 < q ≤ 1) sparse regularization item on the extensively existed noises/outliers is incorporated into the WSTN minimization to enhance its robustness in the impulsive scenarios. Finally, we propose an efficient and scalable robust high-order tensor recovery method solving a double nonconvex optimization with convergence guarantees. Synthetic and real experiments demonstrate that the proposed approach outperforms the state-of-the-art ones in terms of both accuracy and computational complexity.

Recently, tensor Singular Value Decomposition (t-SVD)-based low-rank tensor completion (LRTC) has achieved unprecedented success in addressing various pattern analysis issues. However, existing studies mostly focus on third-order tensors while order- d ( d ≥ 4 ) tensors are commonly encountered in real-world applications, like fourth-order color videos, fourth-order hyper-spectral videos, fifth-order light-field images, and sixth-order bidirectional texture functions. Aiming at addressing this critical issue, this paper establishes an order- d tensor recovery framework including the model, algorithm and theories by innovatively developing a novel algebraic foundation for order- d t-SVD, thereby achieving exact completion for any order- d low t-SVD rank tensors with missing values with an overwhelming probability. Emperical studies on synthetic data and real-world visual data illustrate that compared with other state-of-the-art recovery frameworks, the proposed one achieves highly competitive performance in terms of both qualitative and quantitative metrics. In particular, as the observed data density becomes low, i.e., about 10%, the proposed recovery framework is still significantly better than its peers. The code of our algorithm is released at https://github.com/Qinwenjinswu/TIP-Code.