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Low-rank tensor recovery faces challenges in accurately defining the low-rankness of a tensor. Most existing definitions typically focus on one domain alone — either the original or frequency domain. Additionally, certain definitions often exhibit limitations in their sensitivity to orientation variation. To overcome these challenges, we define a novel tensor rank, the Orientation Invariant Hybrid Rank (OIHR). This rank fuses rank information across all orientations in both frequency and original domains. Employing its convex approximation, the Orientation Invariant Hybrid Nuclear Norm (OIHNN), we propose a general tensor recovery model. We further explore the statistical performance of the estimator based on this model, establishing a deterministic upper bound on the estimation error under generic noise. Furthermore, non-asymptotic upper bounds under Gaussian noise are separately derived for two specific cases: tensor compressive sensing and tensor completion. Finally, we propose the algorithm to solve the model. Extensive experiments on both synthetic and real data are conducted to validate the statistical guarantees and verify the effectiveness of our algorithm. • Our tensor rank fuses low-rankness on all orientations in frequency–original domain. • Tensor recovery algorithm is obtained and outperforms existing convex methods. • Statistical Guarantees are established for estimated error. [ABSTRACT FROM AUTHOR]

Tensor recovery has recently arisen in a lot of application fields, such as transportation, medical imaging, and remote sensing. Under the assumption that signals possess sparse and/or low-rank structures, many tensor recovery methods have been developed to apply various regularization techniques together with the operator-splitting type of algorithms. Due to the unprecedented growth of data, it becomes increasingly desirable to use streamlined algorithms to achieve real-time computation, such as stochastic optimization algorithms that have recently emerged as an efficient family of methods in machine learning. In this work, we propose a novel algorithmic framework based on the Kaczmarz algorithm for tensor recovery. We provide thorough convergence analysis and its applications from the vector case to the tensor one. Numerical results on a variety of tensor recovery applications, including sparse signal recovery, low-rank tensor recovery, image inpainting, and deconvolution, illustrate the enormous potential of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the CP-nuclear value of a completely positive (CP) tensor and study its properties. A semidefinite relaxation algorithm is proposed for solving the minimal CP-nuclear-value tensor recovery. If a partial tensor is CP-recoverable, the algorithm can give a CP tensor recovery with the minimal CP-nuclear value, as well as a CP-nuclear decomposition of the recovered CP tensor. If it is not CP-recoverable, the algorithm can always give a certificate for that, when it is regular. Some numerical experiments are also presented. [ABSTRACT FROM AUTHOR]

Higher-order tensors can represent scores in a rating system, frames in a video, and images of the same subject. In practice, the measurements are often highly quantized due to the sampling strategies or the quality of devices. Existing works on tensor recovery have focused on data losses and random noises. Only a few works consider tensor recovery from quantized measurements but are restricted to binary measurements. This paper, for the first time, addresses the problem of tensor recovery from multi-level quantized measurements by leveraging the low CANDECOMP/PARAFAC (CP) rank property. We study the recovery of both general low-rank tensors and tensors that have tensor singular value decomposition (TSVD) by solving nonconvex optimization problems. We provide the theoretical upper bounds of the recovery error, which diminish to zero when the sizes of dimensions increase to infinity. We further characterize the fundamental limit of any recovery algorithm and show that our recovery error is nearly order-wise optimal. A tensor-based alternating proximal gradient descent algorithm with a convergence guarantee and a TSVD-based projected gradient descent algorithm are proposed to solve the nonconvex problems. Our recovery methods can also handle data losses and do not necessarily need the information of the quantization rule. The methods are validated on synthetic data, image datasets, and music recommender datasets. [ABSTRACT FROM AUTHOR]

• A TNN-based least square estimator is proposed for robust tensor decomposition. We only assume the underlying tensor to satisfy the l ∞ -norm boundedness condition, which is less strict than the tensor incoherence conditions. • On the statistical side, both deterministic and non-asymptotic upper bounds on the estimation error are established in Theorems 1 and 2 , respectively. The non-asymptotic upper bounds are then proved to be minimax near-optimal by Theorem 3. Experiments on synthetic dataset verify that the proposed upper bounds can predict the scaling behavior of the estimation error. • On the computational side, two algorithms, i.e., an ADMM-based algorithm (Algorithm 1) and an FW-based algorithm (Algorithm 1), are proposed with convergence guarantees (Theorems 4 and 5). The latter gets rid of the proximity operator of TNN, and has significantly cheaper one-iteration computational cost. Experiments on real dataset validate the effectiveness and the efficiency of the proposed algorithms. Aiming at recovering a signal tensor from its mixture with outliers and noises, robust tensor decomposition (RTD) arises frequently in many real-world applications. Recently, the low-tubal-rank model has shown more powerful performances than traditional tensor low-rank models in several tensor recovery tasks. Assuming the underlying tensor to be low-tubal-rank and the outliers sparse, this paper first proposes a penalized least squares estimator for RTD. Specifically, we adopt the tubal nuclear norm (TNN) and a sparsity inducing norm to regularize the underlying tensor and the outliers, respectively. Then, from a statistical standpoint, non-asymptotic upper bounds on the estimation error are established and proved to be near-optimal in a minimax sense. Further, two algorithms, namely, an ADMM-based algorithm and a Frank-Wolfe (FW) based algorithm are proposed to efficiently solve the proposed estimator from a computational standpoint. The sharpness of the proposed upper bound is verified on synthetic datasets. The superiority and efficiency of the proposed algorithms is demonstrated in experiments on real datasets. [ABSTRACT FROM AUTHOR]