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- $d$ ( $d\geq 4$ ) 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

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 .

The tensor–tensor product-induced tensor nuclear norm (t-TNN) (Lu et al. , 2020) minimization for low-tubal-rank tensor recovery attracts broad attention recently. However, minimizing the t-TNN faces some drawbacks. For example, the obtained solution could be suboptimal to the original problem due to its loose approximation. In this article, we extract a unified nonconvex surrogate of the tensor tubal rank as a tighter regularizer, which involves many popular nonconvex penalty functions. An iterative reweighted t-TNN algorithm is proposed to solve the resulting generalized nonconvex tubal rank minimization for tensor recovery. It converges to a critical point globally with rigorous proofs based on the Kurdyka–Łojasiwicz property. Furthermore, we provide the theoretical guarantees for exact and robust recovery by developing the tensor null space property. Extensive experiments demonstrate that our approach markedly enhances recovery performance compared with several state-of-the-art convex and nonconvex methods.

Mining structural priors in data is a widely recognized technique for hyperspectral image (HSI) denoising tasks, whose typical ways include model-based methods and data-based methods. The model-based methods have good generalization ability, while the runtime can hardly meet the fast processing requirements of the practical situations due to the large size of an HSI X ϵ R ^MN×B . For the data-based methods, they perform relatively fast on new test data once they have been trained. However, their generalization ability is always insufficient. In this paper, we propose a fast model-based approach via a novel regularizer named the Representative Coefficient Total Variation (RCTV) to simultaneously characterize the low rank and local smooth properties. The RCTV regularizer is proposed based on the observation that the representative coefficient matrix U ϵ R ^MN×R ( R ≪ B ) obtained by orthogonally transforming the original HSI X can inherit the strong local-smooth prior of X. Since R/B is very small, the model based on the RCTV regularizer has lower time complexity. Additionally, we find that the representative coefficient matrix U is robust to noise, and thus the RCTV regularizer can somewhat promote the robustness of the HSI denoising model. Extensive experiments on mixed noise removal demonstrate that the proposed method realizes a perfect compromise between denoising performance and denoising speed as compared with other state-of-the-art methods. Remarkably, the denoising speed of our proposed method outperforms all competing model-based techniques and is comparable with the deep learning-based approaches. The code of our algorithm is released at https://github.com/andrew-pengjj/rctv.git.

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 ( $d\geq 3$ ) 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 recent years, the imputation of spatiotemporal traffic data has emerged as a critical area of research within intelligent transportation systems. A commonly employed approach is low-rank matrix/tensor completion combined with additional spatiotemporal regularization techniques. However, many existing methods simply incorporate these regularizations into the low-rank model, resulting in models that are highly sensitive to trade-off parameters and yield mediocre results. Motivated by this problem, in this paper, we propose a method called Low-Rank Tensor Completion with 3D Spatiotemporal Transform (LRTC-3DST) for traffic data imputation, which effectively integrates low-rankness with three types of spatiotemporal characteristics in a fused manner and thus avoids trade-off parameters. We first design three specific transformations including graph Laplacian transform, fractional difference transform and periodic circulant transform for encoding the spatial local consistency, temporal local consistency and approximate periodicity of 3D traffic data, respectively. Then the LRTC-3DST model is proposed by using the truncated tensor nuclear norm on the three spatiotemporal feature tensors. The proposed model is optimized via the alternating direction multipliers method. Extensive experiments on a series of real traffic datasets demonstrate that our LRTC-3DST significantly outperforms numerous related methods, even when the missing rate is as high as $99\%$ . The code is available at https://github.com/HaoShu2000/LRTC-3DST.