Matrix Nuclear Norm Research Articles

Semi-Proximal ADMM for Primal and Dual Robust Low-Rank Matrix Restoration from Corrupted Observations

The matrix nuclear norm minimization problem has been extensively researched in recent years due to its widespread applications in control design, signal and image restoration, machine learning, big data problems, and more. One popular model is nuclear norm minimization with the l2-norm fidelity term, but it is only effective for those problems with Gaussian noise. A nuclear norm minimization problem with the l1-norm fidelity term has been studied in this paper, which can deal with the problems with not only non-Gaussian noise but also Gaussian noise or their mixture. Moreover, it also keeps the efficiency for the noiseless case. Given the nonsmooth proposed model, we transform it into a separated form by introducing an auxiliary variable and solve it by the semi-proximal alternating direction method of multipliers (sPADMM). Furthermore, we first attempt to solve its dual problem by sPADMM. Then, the convergence guarantees for the aforementioned algorithms are given. Finally, some numerical studies are dedicated to show the robustness of the proposed model and the effectiveness of the presented algorithms.

NMGMDA: a computational model for predicting potential microbe–drug associations based on minimize matrix nuclear norm and graph attention network

The prediction of potential microbe–drug associations is of great value for drug research and development, especially, methods, based on deep learning, have been achieved significant improvement in bio-medicine. In this manuscript, we proposed a novel computational model named NMGMDA based on the nuclear norm minimization and graph attention network to infer latent microbe–drug associations. Firstly, we created a heterogeneous microbe–drug network in NMGMDA by fusing the drug and microbe similarities with the established drug–microbe associations. After this, by using GAT and NNM to calculate the predict scores. Lastly, we created a fivefold cross validation framework to assess the new model NMGMDA's progressiveness. According to the simulation results, NMGMDA outperforms some of the most advanced methods, with a reliable AUC of 0.9946 on both MDAD and aBioflm databases. Furthermore, case studies on Ciprofloxacin, Moxifoxacin, HIV-1 and Mycobacterium tuberculosis were carried out in order to assess the effectiveness of NMGMDA even more. The experimental results demonstrated that, following the removal of known correlations from the database, 16 and 14 medications as well as 19 and 17 microbes in the top 20 predictions were validated by pertinent literature. This demonstrates the potential of our new model, NMGMDA, to reach acceptable prediction performance.

Non-uniformly sampled 2D NMR Spectroscopy reconstruction based on Low Rank Hankel Matrix Fast Tri-Factorization and Non-convex Factorization

Nuclear magnetic resonance (NMR) spectroscopy is widely used in chemistry and medicine to research the composition of matter and the spatial structure of proteins. However, NMR signals with larger sizes and higher dimensions are slower to acquire. Therefore, Non-Uniform Sampling (NUS) methods are used to speed up the acquisition of NMR signals, and some mathematical algorithms are used to recover the original NMR signals from the NUS data. In recent years, low-rank Hankel matrix completion is proved to recover NMR data with less error. However, it requires singular value decomposition (SVD) with high computational complexity in each iteration, which is not suitable for reconstructing signals with large sizes. In this paper, we propose a Fast Tri-Factorization (FTF) method to decompose the low-rank Hankel matrix into three small-scale matrices, and it convert the original problem into a matrix nuclear norm minimization problem on the small-scale matrix to mitigate high computation cost of multiple SVDs. Furthermore, the factor matrix size remains constant for different size of the Hankel matrices. We further replace the low-rank constraint with Non-convex Factorization (NF) to avoid the time-consuming SVD in each iteration. Numerical experiments show that the proposed method can significantly speed up the NMR reconstruction compared with several existing algorithms while ensuring the reconstruction accuracy.

A robust mixed error coding method based on nonconvex sparse representation

Linear representation based methods have been extensively applied in image recognition, especially for those with noise, illumination changes, and occlusions. However, most existing methods assume a specific distribution for image noise estimation, which is intractable to handle complex variations. Besides, they usually use convex norm to describe the noise sparse and low-rank property, and it is a biased approximation. To address these problems, we propose a novel nonconvex regularized robust mixed error coding (NRRM) method, which uses mixed norms from both 1D and 2D perspectives to model the complex image noise without convex relaxation. In specific, we use weighted ℓ2-norm based robust coding to characterize the sparse noise in images, and weighted matrix nuclear norm to characterize the low-rank noise. Compared with traditional regression approaches, our method can more fine-grained and accurate to capture noise and alleviate its negative influence for robust recognition. Besides, we constrain the representation component in a group-wise manner to weigh the roles of different classes. The NRRM model is solved efficiently by adopting an alternating direction method of multipliers (ADMM) algorithm. Comprehensive experiments on some benchmark face image databases validate the superiority of NRRM over several state-of-the-art linear representation based methods.

Effective Incomplete Multi-View Clustering via Low-Rank Graph Tensor Completion

In the past decade, multi-view clustering has received a lot of attention due to the popularity of multi-view data. However, not all samples can be observed from every view due to some unavoidable factors, resulting in the incomplete multi-view clustering (IMC) problem. Up until now, most efforts for the IMC problem have been made on the learning of consensus representations or graphs, while many missing views are ignored, making it impossible to capture the information hidden in the missing view. To overcome this drawback, we first analyzed the low-rank relationship existing inside each graph and among all graphs, and then propose a novel method for the IMC problem via low-rank graph tensor completion. Specifically, we first stack all similarity graphs into a third-order graph tensor and then exploit the low-rank relationship from each mode using the matrix nuclear norm. In this way, the connection hidden between the missing and available instances can be recovered. The consensus representation can be learned from all completed graphs via multi-view spectral clustering. To obtain the optimal multi-view clustering result, incomplete graph recovery and consensus representation learning are integrated into a joint framework for optimization. Extensive experimental results on several incomplete multi-view datasets demonstrate that the proposed method can obtain a better clustering performance in comparison with state-of-the-art incomplete multi-view clustering methods.

MNNMDA: Predicting human microbe-disease association via a method to minimize matrix nuclear norm

Identifying the potential associations between microbes and diseases is the first step for revealing the pathological mechanisms of microbe-associated diseases. However, traditional culture-based microbial experiments are expensive and time-consuming. Thus, it is critical to prioritize disease-associated microbes by computational methods for further experimental validation. In this study, we proposed a novel method called MNNMDA, to predict microbe-disease associations (MDAs) by applying a Matrix Nuclear Norm method into known microbe and disease data. Specifically, we first calculated Gaussian interaction profile kernel similarity and functional similarity for diseases and microbes. Then we constructed a heterogeneous information network by combining the integrated disease similarity network, the integrated microbe similarity network and the known microbe-disease bipartite network. Finally, we formulated the microbe-disease association prediction problem as a low-rank matrix completion problem, which was solved by minimizing the nuclear norm of a matrix with a few regularization terms. We tested the performances of MNNMDA in three datasets including HMDAD, Disbiome, and Combined Data with small, medium and large sizes respectively. We also compared MNNMDA with 5 state-of-the-art methods including KATZHMDA, LRLSHMDA, NTSHMDA, GATMDA, and KGNMDA, respectively. MNNMDA achieved area under the ROC curves (AUROC) of 0.9536 and 0.9364 respectively on HDMAD and Disbiome, better than the AUCs of compared methods under the 5-fold cross-validation for all microbe-disease associations. It also obtained a relatively good performance with AUROC 0.8858 in the combined data. In addition, MNNMDA was also better than other methods in area under precision and recall curve (AUPR) under the 5-fold cross-validation for all associations, and in both AUROC and AUPR under the 5-fold cross-validation for diseases and the 5-fold cross-validation for microbes. Finally, the case studies on colon cancer and inflammatory bowel disease (IBD) also validated the effectiveness of MNNMDA. In conclusion, MNNMDA is an effective method in predicting microbe-disease associations. AvailabilityThe codes and data for this paper are freely available at Github https://github.com/Haiyan-Liu666/MNNMDA.

Drug\u2013target interaction prediction using unifying of graph regularized nuclear norm with bilinear factorization

BackgroundWet-lab experiments for identification of interactions between drugs and target proteins are time-consuming, costly and labor-intensive. The use of computational prediction of drug–target interactions (DTIs), which is one of the significant points in drug discovery, has been considered by many researchers in recent years. It also reduces the search space of interactions by proposing potential interaction candidates.ResultsIn this paper, a new approach based on unifying matrix factorization and nuclear norm minimization is proposed to find a low-rank interaction. In this combined method, to solve the low-rank matrix approximation, the terms in the DTI problem are used in such a way that the nuclear norm regularized problem is optimized by a bilinear factorization based on Rank-Restricted Soft Singular Value Decomposition (RRSSVD). In the proposed method, adjacencies between drugs and targets are encoded by graphs. Drug–target interaction, drug-drug similarity, target-target, and combination of similarities have also been used as input.ConclusionsThe proposed method is evaluated on four benchmark datasets known as Enzymes (E), Ion channels (ICs), G protein-coupled receptors (GPCRs) and nuclear receptors (NRs) based on AUC, AUPR, and time measure. The results show an improvement in the performance of the proposed method compared to the state-of-the-art techniques.

Tensor Recovery via $*_L$-Spectral $k$-Support Norm

Unlike traditional tensor decompositions which model low-rankness in the original domain, the recently proposed tensor <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -Singular Value Decomposition ( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -SVD) casts a new light on tensor analysis by exploiting low-rankness in the spectral domain. For convexized rank minimization within the framework of <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -SVD, the tensor <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -Tubal Nuclear Norm ( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -TNN) out-performs classical tensorial extensions of matrix nuclear norm in many applications. In this paper, we first generalize <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -TNN to the <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> *L</sub> -Spectral k-Support Norm ( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -SpSN-k)as a new low-rank regularizer, and then adopt it to formulate two estimators for tensor recovery from noisy linear observations. Further, statistical performance of the proposed estimators is analyzed by establishing both deterministic and non-asymptotic upper bounds on the estimation error, which indicates that the proposed tensor Dantzig selector enjoys near-optimal estimation error for tensor compressive sensing and tensor completion. Moreover, proximal operator of the proposed norm is derived and embeded in an efficient algorithm to compute the estimators. Experiments on synthetic datasets verify correctness of the proposed error bounds and image inpainting results demonstrate superiority of the proposed estimators.

Repositioning Drugs on Human Influenza A Viruses Based on a Novel Nuclear Norm Minimization Method.

Influenza A viruses, especially H3N2 and H1N1 subtypes, are viruses that often spread among humans and cause influenza pandemic. There have been several big influenza pandemics that have caused millions of human deaths in history, and the threat of influenza viruses to public health is still serious nowadays due to the frequent antigenic drift and antigenic shift events. However, only few effective anti-flu drugs have been developed to date. The high development cost, long research and development time, and drug side effects are the major bottlenecks, which could be relieved by drug repositioning. In this study, we proposed a novel antiviral Drug Repositioning method based on minimizing Matrix Nuclear Norm (DRMNN). Specifically, a virus-drug correlation database consisting of 34 viruses and 205 antiviral drugs was first curated from public databases and published literature. Together with drug similarity on chemical structure and virus sequence similarity, we formulated the drug repositioning problem as a low-rank matrix completion problem, which was solved by minimizing the nuclear norm of a matrix with a few regularization terms. DRMNN was compared with three recent association prediction algorithms. The AUC of DRMNN in the global fivefold cross-validation (fivefold CV) is 0.8661, and the AUC in the local leave-one-out cross-validation (LOOCV) is 0.6929. Experiments have shown that DRMNN is better than other algorithms in predicting which drugs are effective against influenza A virus. With H3N2 as an example, 10 drugs most likely to be effective against H3N2 viruses were listed, among which six drugs were reported, in other literature, to have some effect on the viruses. The protein docking experiments between the chemical structure of the prioritized drugs and viral hemagglutinin protein also provided evidence for the potential of the predicted drugs for the treatment of influenza.

Hyperspectral-Multispectral Image Fusion via Tensor Ring and Subspace Decompositions

Fusion from a spatially low resolution hyperspectral image (LR-HSI) and a spectrally low resolution multispectral image (MSI) to produce a high spatial-spectral HSI (HR-HSI), known as hyperspectral super resolution, has risen to a preferred topic for reinforcing the spatial-spectral resolution of HSI in recent years. In this work, we propose a new model, namely, low-rank tensor ring decomposition based on tensor nuclear norm (LRTRTNN), for HSI-MSI fusion. Specifically, for each spectrally subspace cube, similar patches are grouped to exploit both the global low-rank property of LR-HSI and the nonlocal similarity of HR-MSI. Afterward, a joint optimization of all groups via the presented LRTRTNN approximation is implemented in a unified cost function. With the introduced tensor nuclear norm (TNN) constraint, all 3D tensor ring factors are no longer unfolded to suit the matrix nuclear norm used in conventional methods, and the internal tensor structure can be naturally retained. The alternating direction method of multipliers is introduced for coefficients update. Numerical and visual experiments on real data show that our LRTRTNN method outperforms most state-of-the-art algorithms in terms of fusing performance.

Weighted Tensor Low-Rankness and Learnable Analysis Sparse Representation Model for Texture Preserving Low-Dose CT Reconstruction

In CT images, tissue structures and lesion changes illustrate evident non-local self-similarity and regionally constant properties. The low-rank model and the learnable sparse representation model are powerful tools that can respectively encode the correlations among non-local similar patches and the sparsity in a local transformed subspace about the underlying CT image. Existing Model-Based Iterative Reconstruction (MBIR) methods generally adopt one of the two models alone for CT reconstruction, which might suffer from modelling deficiency and hampers their reconstruction performance. In this study, we presented a novel Weighted Tensor Low-Rank and Learnable Analysis Sparse Representation model (WTLR-LASR) to simultaneously encode the non-local correlations and local transformed sparsity natures. Specifically, we developed a novel Weighted Tensor Nuclear Norm Minimization (WTNNM) formulation to characterize the weighted tensor low-rank model, and introduced the Weighted Tensor Nuclear Norm Proximal (WTNNP) operator to solve the non-convex WTNNM problem. We further proved that the WTNNP problem can be equivalently transformed to a weighted matrix nuclear norm proximal (WMNNP) problem in the Fourier transform domain, which allowed us to easily reach the closed-form optimum of the WTNNP problem. We proposed a novel CT reconstruction algorithm based on the presented WTLR-LASR model. We also introduced a genetic algorithm to automatically select the parameters in the proposed algorithm. Extensive experimental studies were performed to validate the effectiveness of the proposed algorithm. The results demonstrate that the proposed algorithm can achieve noticeable improvements over state-of-the-art methods in terms of noise suppression and textures preservation.

Improved robust tensor principal component analysis for accelerating dynamic MR imaging reconstruction.

Dynamic magnetic resonance imaging (dMRI) strikes a balance between reconstruction speed and image accuracy in medical imaging field. In this paper, an improved robust tensor principal component analysis (RTPCA) method is proposed to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data. The MR reconstruction problem is formulated as a high-order low-rank tenor plus sparse tensor recovery problem, which is solved by robust tensor principal component analysis (RTPCA) with a new tensor nuclear norm (TNN). To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into TNN for enforcing the low-rank structure in MRI datasets. The experimental results show that the proposed method outperforms state-of-the-art methods in terms of both MR image reconstruction accuracy and computational efficiency on 3D and 4D experiment datasets, especially for 4D MR image reconstruction. Graphical abstract The flowchart of the proposed method to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data in the kth iteration. To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into tensor nuclear norm (TNN) for enforcing the low-rank structure in MRI datasets. In each iteration, the first step is to get low-rank tensor ℓk - 1 by using soft thresholding on the singular values of ℓk - 1 = χk - 1 - ξk - 1, and an improved tensor nuclear norm method is proposed to process the low-rank tensor ℓk - 1 firstly. Then, the shrinkage operator is applied to ξk - 1 = χk - 1 - ℓk - 1 for sparse part ξk - 1. The final reconstructed d-MRI χk is obtained by enforcing data consistency that the residual in K-space is subtracted by the sum of the reconstructed low-rank tensor and sparse tensor.

A Global Linear and Local Superlinear (Quadratic) Inexact Non-Interior Continuation Method for Variational Inequalities Over General Closed Convex Sets

We use the concept of barrier-based smoothing approximations to extend the non-interior continuation method, which was proposed by B. Chen and N. Xiu for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions, to an inexact non-interior continuation method for variational inequalities over general closed convex sets. Newton equations involved in the method are solved inexactly to deal with high dimension problems. The method is proved to have global linear and local superlinear/quadratic convergence under suitable assumptions. We apply the method to non-negative orthants, positive semidefinite cones, polyhedral sets, epigraphs of matrix operator norm cone and epigraphs of matrix nuclear norm cone.

Robust Schatten-p Norm Based Approach for Tensor Completion

The matrix nuclear norm has been widely applied to approximate the matrix rank for low-rank tensor completion because of its convexity. However, this relaxation may make the solution seriously deviate from the original solution for real-world data recovery. In this paper, using a nonconvex approximation of rank, i.e., the Schatten-p norm, we propose a novel model for tensor completion. It’s hard to solve this model directly because the objective function of the model is nonconvex. To solve the model, we develop a variant of this model via the classical quadric penalty method, and propose an algorithm, i.e., SpBCD, based on the block coordinate descent method. Although the objective function of the variant is nonconvex, we show that the sequence generated by SpBCD is convergent to a critical point. Our numerical experiments on real-world data show that SpBCD delivers state-of-art performance in recovering missing data.

The non-convex geometry of low-rank matrix optimization

AbstractThis work considers two popular minimization problems: (i) the minimization of a general convex function f(X) with the domain being positive semi-definite matrices, and (ii) the minimization of a general convex function f(X) regularized by the matrix nuclear norm $\|X\|_{*}$ with the domain being general matrices. Despite their optimal statistical performance in the literature, these two optimization problems have a high computational complexity even when solved using tailored fast convex solvers. To develop faster and more scalable algorithms, we follow the proposal of Burer and Monteiro to factor the low-rank variable $X = UU^{\top } $ (for semi-definite matrices) or $X=UV^{\top } $ (for general matrices) and also replace the nuclear norm $\|X\|_{*}$ with $\big(\|U\|_{F}^{2}+\|V\|_{F}^{2}\big)/2$. In spite of the non-convexity of the resulting factored formulations, we prove that each critical point either corresponds to the global optimum of the original convex problems or is a strict saddle where the Hessian matrix has a strictly negative eigenvalue. Such a nice geometric structure of the factored formulations allows many local-search algorithms to find a global optimizer even with random initializations.

An Improved Algorithm to Enhance Recovery Stability for Low-Rank Image

When recovering original image for low-rank image with noise, the effect usually has not been satisfactory through minimizing matrix nuclear norm to obtain low-rank resolution. Aiming at this problem, we introduced Frobenius norm of low-rank matrix, and combined with original low-rank matrix nuclear nor to form new regular item, and utilized an augmented lagrange multiplier method to resolve the problem after convex relaxation. Through adding 2 F A item on the base of original low-rank image recovery model, we can get better denoising result for low-rank image. The experimental results indicated that, by selecting proper parameters, the improved algorithm has superior recovery result compared to traditional LRMR model on wiping off impulse noise and gaussian noise.

Hankel Matrix Nuclear Norm Regularized Tensor Completion for $N$-dimensional Exponential Signals

Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology, and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired, and thus, how to recover the full signal becomes an active research topic, but existing approaches cannot efficiently recover <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -dimensional exponential signals with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N\geq 3$</tex-math></inline-formula> . In this paper, we study the problem of recovering <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -dimensional (particularly <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N\geq 3$</tex-math></inline-formula> ) exponential signals from partial observations, and formulate this problem as a low-rank tensor completion problem with exponential factor vectors. The full signal is reconstructed by simultaneously exploiting the CANDECOMP/PARAFAC tensor structure and the exponential structure of the associated factor vectors. The latter is promoted by minimizing an objective function involving the nuclear norm of Hankel matrices. Experimental results on simulated and real magnetic resonance spectroscopy data show that the proposed approach can successfully recover full signals from very limited samples and is robust to the estimated tensor rank.

The Twist Tensor Nuclear Norm for Video Completion.

In this paper, we propose a new low-rank tensor model based on the circulant algebra, namely, twist tensor nuclear norm (t-TNN). The twist tensor denotes a three-way tensor representation to laterally store 2-D data slices in order. On one hand, t-TNN convexly relaxes the tensor multirank of the twist tensor in the Fourier domain, which allows an efficient computation using fast Fourier transform. On the other, t-TNN is equal to the nuclear norm of block circulant matricization of the twist tensor in the original domain, which extends the traditional matrix nuclear norm in a block circulant way. We test the t-TNN model on a video completion application that aims to fill missing values and the experiment results validate its effectiveness, especially when dealing with video recorded by a nonstationary panning camera. The block circulant matricization of the twist tensor can be transformed into a circulant block representation with nuclear norm invariance. This representation, after transformation, exploits the horizontal translation relationship between the frames in a video, and endows the t-TNN model with a more powerful ability to reconstruct panning videos than the existing state-of-the-art low-rank models.

Atom Decomposition Based Subgradient Descent for matrix classification

Matrices are appropriate for representing a wealth of data with complex structures such as images and electroencephalogram data (EEG). To learn a classifier dealing with these matrix data, the structure information of the feature matrix is useful. In this paper, we focus on the regularized matrix classifiers whose input samples and weight parameters are both in the form of a matrix. Some existing approaches assume that the weight matrix has a low-rank structure and then utilize the popular nuclear norm of the weight matrix as a regularization term. However, the optimization methods for these matrix classifiers often involve numbers of expensive singular value decomposition (SVD) operations, which prevents scaling beyond moderate matrix sizes. To reduce the time complexity, we propose a novel learning algorithm called Atom Decomposition Based Subgradient Descent (ADBSD), which solves the optimization problem for the matrix classifier whose objective function is the combination of the Frobenius matrix norm and nuclear norm of the weight matrix along with the hinge loss function. Our ADBSD is an iterative scheme which selects the most informative rank-one matrices from the subgradient of the objective function in each iteration. We consider using the atom decomposition based methods to minimize nuclear norm because they mainly rely on the computation of top singular vector pair which leads to great advantages on efficiency. We empirically evaluate the performance of the proposed algorithm ADBSD on both synthetic and real-world datasets. Results show that our approach is more efficient and robust than the state-of-the-art methods.

Rank-1 Tensor Properties with Applications to a Class of Tensor Optimization Problems

This paper studies models and algorithms for a class of tensor optimization problems, based on a rank-1 equivalence property between a tensor and certain unfoldings. It is first shown that in $d$th order tensor space, the set of rank-1 tensors is the same as the intersection of $\lceil \log_2(d) \rceil$ tensor sets, of which tensors have a specific rank-1 balanced unfolding matrix. Moreover, the number $\lceil \log_2(d) \rceil$ is proved to be optimal in some sense. Based on the above equivalence property, three relaxation approaches for solving the best rank-1 tensor approximation problems are proposed, including two convex relaxations and a nonconvex one. The two convex relaxations utilize the matrix nuclear norm regularization/constraints. They have the advantage of identifying whether the solution is a global optimizer of the original problem, by computing the nuclear norm or the Frobenius norm of a certain matrix. Under certain assumptions, the optimal solution of the original problem is characterize...