Schatten P-norm Research Articles

Commutator bounds and region of singular values of the commutator with a rank one matrix

The general problem under consideration is the determination of the best (i.e., smallest) constant Cp,q,r such that‖XY−YX‖p≤Cp,q,r‖X‖q‖Y‖rfor all n×n matrices X and Y, where ‖⋅‖p is the Schatten p-norm. Among the open situations, the problem is solved when(i)2<p<∞, q=r=1;(ii)2<p<∞, q=1, 1<r<2, and X, Y are 2×2 real matrices. The result for (i) is obtained via the study of the region (recently obtained in [5]) of singular values of the commutator XY−YX, where X and Y are normalized rank one matrices. As a result of independent interest, for 2×2 real matrices, the region of singular values of the commutator XY−YX, where X is rank one and normalized, and ‖Y‖r=1, is determined. The result for (ii) is a consequence of the case when r lies between 1 and 2.

Multiple kernel low-rank representation-based robust multi-view subspace clustering

Owing to the presence of complex noise, it is extremely challenging to learn a low-dimensional subspace structure directly from the original data. In addition, the nonlinear structure of the data makes multi-view subspace clustering more difficult. In this paper, we propose a multiple kernel low-rank representation-based robust multi-view subspace clustering method (MKLR-RMSC) that combines a learnable low-rank multiple kernel trick with co-regularization. MKLR-RMSC mainly conducts the following four tasks: 1) fully mining the complementary information provided by the different views in the feature spaces, 2) the containment of multiple low-dimensional subspaces in the feature space data, 3) allowing all view-specific representations towards a common centroid, and 4) effectively dealing with non-Gaussian noise in data. In our model, the weighted Schatten p-norm is applied to fully explore the effects of different ranks while approaching the original low-rank hypothesis. Moreover, different predefined learning kernel matrices are designed for different views, which is more conducive to mining the unique and complementary information of different views. In addition, as a robust measure, correntropy is applied in MKLR-RMSC. Our method is more effective and robust than several of the most advanced methods on six commonly used datasets.

Sparse-view CBCT reconstruction via weighted Schatten p-norm minimization.

A novel iterative algorithm is proposed for sparse-view cone beam computed tomography (CBCT) reconstruction based on the weighted Schatten p-norm minimization (WSNM). By using the half quadratic splitting, the sparse-view CBCT reconstruction task is decomposed into two sub-problems that can be solved through alternating iteration: simple reconstruction and image denoising. The WSNM that fits well with the low-rank hypothesis of CBCT data is introduced to improve the denoising sub-problem as a regularization term. The experimental results based on the digital brain phantom and clinical CT data indicated the advantages of the proposed algorithm in both structural information preservation and artifacts suppression, which performs better than the classical algorithms in quantitative and qualitative evaluations.

Robust energy preserving embedding for multi-view subspace clustering

With the increasing popularity of multi-view data, multi-view subspace clustering (MVSC) has attracted intensive attention. However, the curse of dimensionality of the high-dimensional multi-view data constantly obsesses the development of multi-view method. Together with the curse of dimensionality, the existence of noise and outliers further obstructs the ability of MVSC to exploit the underlying subspace structure. To solve these challenging problems, in this paper, we propose a novel MVSC method, namely robust energy preserving embedding (REPE), which aims to exploit the underlying subspace structure of multi-view data through energy preserving embedding projection and low-rank constraint. In a different manner than before, we use dictionary learning to integrate self-expressiveness learning and embedded subspace learning together, ensure the preservation of energy by restoring the data in embedding space to the original space. Moreover, Schatten p-norm is introduced to better approximate the low-rank constraint. So that we can reduce the dimension of the data while maintain the major energy of the original view data. Thereby, a high quality affinity graph can be learned, such that the promising clustering results can subsequently be obtained. Extensive experiments on five benchmarks have validated the effectiveness of the proposed method, by comparing it with the state of arts.

Graph-structure constraint and Schatten p-norm-based unsupervised domain adaptation for image classification

Unsupervised domain adaptation, which aims to classify a target domain correctly only using a labeled source domain, has achieved promising performance yet remains a challenging problem. Most traditional methods focus on exploiting either geometric or statistical characteristics to reduce domain shifts. To take advantage of both sides, in this paper, we propose a unified framework incorporating both the geometric and statistical characteristics by adopting the non-convex Schatten p-norm and graph Laplacian constraints to preserve global and local structure information and constructing marginal and conditional distribution minimization terms to reduce the distribution shifts. Moreover, a classification error term on the source domain is embedded into the objective function to increase the discriminability. The proposed method has been evaluated on six datasets and the experimental results demonstrate the superiority of the proposed method over several state-of-the-art methods. The MATLAB code of our method will be publicly available at https://github.com/HeyouChang/unsupervised-domain-adaptation .

3D MR image denoising using a modified adaptive high order singular value decomposition method.

Magnetic resonance (MR) images are generally degraded by random noise governed by Rician distributions. In this study, we developed a modified adaptive high order singular value decomposition (HOSVD) method, taking consideration of the nonlocal self-similarity and weighted Schatten p-norm. We extracted 3D cubes from noise images and classified the similar cubes by the Euclidean distance between cubes to construction a fourth-order tensor. Each rank of unfolding matrices was adaptively determined by weighted Schatten p-norm regularization. The latent noise-free 3D MR images can be obtained by an adaptive HOSVD. Denoising experiments were tested on both synthetic and clinical 3D MR images, and the results showed the proposed method outperformed several existing methods for Rician noise removal in 3D MR images.

Multi-band weighted [formula omitted] norm minimization for image denoising

Low rank matrix approximation (LRMA) has a wide range of applications in computer vision and has drawn much attention in recent years. A typical nuclear norm minimization (NNM) is often used to solve a LRMA, but this is likely to overshrink the rank components due to having the same threshold. To address this problem, we propose a flexible and precise model named multi-band weighted lp norm minimization (MBWPNM). We have reformulated it into nonconvex lp norm subproblems under certain weight conditions, and we solve these subproblems via a generalized soft-thresholding algorithm. We then adopt MBWPNM for image (grayscale, color and multispectral) denoising. The proposed MBWPNM not only guarantees a more accurate approximation with a Schatten p-norm in case of a change of noise levels, but it also considers prior knowledge, for which different rank components have varying importance. Extensive experiments on additive white Gaussian noise removal and realistic noise removal demonstrate that the proposed MBWPNM achieves better performance than several state-of-the-art algorithms.

Blind Image Deblurring via the Weighted Schatten p-norm Minimization Prior

In this paper, we propose a new image blind deblurring model, based on a novel low-rank prior. As the low-rank prior, we employ the weighted Schatten p-norm minimization (WSNM), which can represent both the sparsity and self-similarity of the image structure more accurately. In addition, the L0-regularized gradient prior is introduced into our model, to extract significant edges quickly and effectively. Moreover, the WSNM prior can effectively eliminate harmful details and maintain dominant edges, to generate sharper intermediate images, which is beneficial for blur kernel estimation. To optimize the model, an efficient optimization algorithm is developed by combining the half-quadratic splitting strategy with the generalized soft-thresholding algorithm. Extensive experiments have demonstrated the validity of the WSNM prior. Our flexible low-rank prior enables the proposed algorithm to achieve excellent results in various special scenarios, such as the deblurring of text, face, saturated, and noise-containing images. In addition, our method can be extended naturally to non-uniform deblurring. Quantitative and qualitative experimental evaluations indicate that the proposed algorithm is robust and performs favorably against state-of-the-art algorithms.

A Non-convex Optimization Model for Signal Recovery

The electroencephalogram (EEG) signal is one of the most frequently used biomedical signals. In order to accurately exploit the cosparsity and low-rank property which is nature in multichannel EEG signals, motivated by the fact that weighted schatten-p norm and $${l_q}$$ norm can better approximate the matrix rank and $${l_0}$$ norm, in this paper, a non-convex optimization model is proposed to precisely reconstruct the multichannel EEG signal. weighted schatten-p norm and $${l_q}$$ norm are used to enforce low-rank property and cosparsity. In addition, an efficient iterative optimization method based on alternating direction method of multipliers is used to solve the resulting non-convex optimization problem. Experimental results have demonstrated that the proposed algorithm can significantly outperform existing state-of-the-art CS methods for compressive sensing of multichannel EEG signals.

Fast algorithm for large‐scale subspace clustering by LRR

Low-rank representation (LRR) and its variants have been proved to be powerful tools for handling subspace clustering problems. Most of these methods involve a sub-problem of computing the singular value decomposition of an n × n matrix, which leads to a computation complexity of O ( n 3 ) . Obviously, when n is large, it will be time consuming. To address this problem, the authors introduce a fast solution, which reformulates the large-scale problem to an equal form with smaller size. Thus, the proposed method remarkably reduces the computation complexity by solving a small-scale problem. Theoretical analysis proves the efficiency of the proposed model. Furthermore, we extend LRR to a general model by using Schatten p-norm instead of nuclear norm and present a fast algorithm to solve large-scale problem. Experiments on MNIST and Caltech101 databse illustrate the equivalence of the proposed algorithm and the original LRR solver. Experimental results show that the proposed algorithm is remarkably faster than traditional LRR algorithm, especially in the case of large sample number.

Blind image recovery approach combing sparse and low-rank regularizations

The only useful prior knowledge in blind compressive sensing is that a signal is sparse in an unknown dictionary. Usually, general dictionaries cannot sparsify all images well. It simultaneously optimizes the dictionary and sparse coefficient in the reconstruction process and has been demonstrated to obtain same results as those compressive sensing techniques based on the known dictionary. In this paper, we propose a novel blind compressive sensing method combing sparse and low-rank regularizations to obtain competitive recovery results. We employ truncated Schatten-p norm and lq norm to approximate rank and norm. At last, we give an optimization strategy based on alternating direction method of multipliers to solve the recovery model. Experimental results prove that our approach could obtain the higher Peak Signal to Noise Ratio values than other competitive methods.

Inequalities for accretive-dissipative block matrices involving convex and concave functions

We present some unitarily invariant norm and Schatten p-norm inequalities relevant to accretive-dissipative matrices involving convex and concave functions. In particular, we show that if is an accretive-dissipative block matrix with , , and f is an increasing convex function on with , then for every unitarily invariant norm . We also extract several inequalities for accretive-dissipative operator matrices. The obtained inequalities extend some known results.

Urban network-wide traffic speed estimation with massive ride-sourcing GPS traces

The ability to obtain accurate estimates of city-wide urban traffic patterns is essential for the development of effective intelligent transportation systems and the efficient operation of smart mobility platforms. This paper focuses on the network-wide traffic speed estimation, using trajectory data generated by a city-wide fleet of ride-sourcing vehicles equipped with GPS-capable smartphones. A cell-based map-matching technique is proposed to link vehicle trajectories with road geometries, and to produce network-wide spatio-temporal speed matrices. Data limitations are addressed using the Schatten p-norm matrix completion algorithm, which can minimize speed estimation errors even with high rates of data unavailability. A case study using data from Chengdu, China, demonstrates that the algorithm performs well even in situations involving continuous data loss over a few hours, and consequently, addresses large-scale network-wide traffic state estimation problems with missing data, while at the same time outperforming other data recovery techniques that were used as benchmarks. Our approach can be used to generate congestion maps that can help monitor and visualize traffic dynamics across the network, and therefore form the basis for new traffic management, proactive congestion identification, and congestion mitigation strategies.

Weighted t-Schatten-p Norm Minimization for Real Color Image Denoising

In this paper, to fully exploit the spatial and spectral correlation information, we present a new real color image denoising scheme using tensor Schatten-p norm (t-Schatten-p norm) minimization based on t-SVD to recover the underlying low-rank tensor. Similar to matrix Schatten-p norm, using non-convex t-Schatten-p (0 <; p <; 1) norm minimization could obtain better results than the tensor nuclear norm minimization which is a convex relaxation of the nonconvex tensor tubal rank. To avoid over-shrink the tensor tubal rank components, a flexible weighted t-Schatten-p norm model is proposed with weights assigned to different elements of tensor singular tubes. We adopt the generalized iterated shrinkage algorithm to solve the minimization problem efficiently. Extensive experiments on one synthetic and two realistic datasets demonstrate the effectiveness of our proposed method to remove noise both quantitatively and qualitatively.

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.

Robust Image Feature Extraction via Approximate Orthogonal Low-Rank Embedding

Feature extraction (FE) plays an important role in machine learning. In order to handle the “dimensionality disaster”problem, the usual approach is to transform the original sample into a low-dimensional target space, in which the FE task is performed. However, the data in reality will always be corrupted by various noises or interfered by outliers, making the task of feature extraction extremely challenging. Thence, we propose a novel image FE method via approximate orthogonal low-rank embedding (AOLRE), which adopts an orthogonal matrix to reserve the major energy of the samples, and the introduction of the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm makes the features more compact, differentiated and interpretable. In addition, the weighted Schatten p-norm is adopted in this model for fully exploring the effects of different ranks while approaching the original hypothesis of low rank. Meanwhile, as a robust measure, the correntropy is applied in AOLRE. This can effectively suppress the adverse influences of contaminated data and enhance the robustness of the algorithm. Finally, the introduction of the classification loss item allows our model to effectively fit the supervised scene. Five common datasets are used to evaluate the performance of AOLRE. The results show that the recognition accuracy and robustness of AOLRE are significantly better than those of several advanced FE algorithms, and the improvement rate ranges from 2% to 15%.

Group-Based Sparse Representation Based on lp -Norm Minimization for Image Inpainting

As a powerful statistical image modeling technique, sparse representation has been successfully applied in various image restoration applications. Most traditional methods depend on ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm optimization and patch-based sparse representation models. However, these methods have two limits: high computational complexity and the lack of the relationship among patches. To solve the above problems, we choose the group-based sparse representation models to simplify the computing process and realize the nonlocal self-similarity of images by designing the adaptive dictionary. Meanwhile, we utilize Ipnorm minimization to solve nonconvex optimization problems based on the weighted Schatten p-norm minimization, which can make the optimization model more flexible. Experimental results on image inpainting show that the proposed method has a better performance than many current state-of-the-art schemes, which are based on the pixel, patch, and group respectively, in both peak signal-to-noise ratio and visual perception.

Majorized Proximal Alternating Imputation for regularized rank constrained matrix completion

Low rank approximation in the presence of missing data is ubiquitous in many areas of science and engineering. As an NP-hard problem, spectral regularization (e.g., nuclear norm, Schatten p-norm) is widely used. However, solving this problem directly can be computationally expensive due to the unknown rank of variables and full rank Singular Value Decompositions (SVD). Bilinear factorization is efficient and effectively benefits from the fast numerical optimization. However, most of existing methods require explicit knowledge of the rank and often tend to be flatlining or over fitting. To balance and take advantage of both, we formulate the matrix completion problem under the regularized rank constrained conditions. We relax this problem to the surrogates with Majorization Minimization (MM). By employing a proximal alternating minimization method to optimize the corresponding surrogates, we propose an efficient and accurate algorithm named Majorized Proximal Alternating Imputation (MPA-IMPUTE) algorithm. The proposed method iteratively replaces the missing data at each step with the most recent estimate, and only involves simple matrix computation and small scale SVD. Theoretically, the proposed method is guaranteed to converge to the stationary point under mild conditions. Moreover, the proposed method can be generalized to a family of non-convex regularization problems. Extensive experiments over synthetic data and real-world dataset demonstrate the superior performance of our method.

Low-Rank Matrix Recovery via Modified Schatten-p Norm Minimization with Convergence Guarantees.

In recent years, low-rank matrix recovery problems have attracted much attention in computer vision and machine learning. The corresponding rank minimization problems are both combinational and NP-hard in general, which are mainly solved by both nuclear norm and Schatten-p (0<p<1) norm based optimization algorithms. However, inspired by weighted nuclear norm and Schatten-p norm as the relaxations of rank function, the main merits of this work firstly provide a modified Schatten-p norm in the affine matrix rank minimization problem, denoted as the modified Schatten-p norm minimization (MSpNM). Secondly, its surrogate function is constructed and the equivalence relationship with the MSpNM is further achieved. Thirdly, the iterative singular value thresholding algorithm (ISVTA) is devised to optimize it, and its accelerated version, i.e., AISVTA, is also obtained to reduce the number of iterations through the well-known Nesterov's acceleration strategy. Most importantly, the convergence guarantees and their relationship with objective function, stationary point and variable sequence generated by the proposed algorithms are established under some specific assumptions, e.g., Kurdyka-Łojasiewicz (KŁ) property. Finally, numerical experiments demonstrate the effectiveness of the proposed algorithms in the matrix completion problem for image inpainting and recommender systems. It should be noted that the accelerated algorithm has a much faster convergence speed and a very close recovery precision when comparing with the proposed non-accelerated one.

Robust Image Compressive Sensing Based on Truncated Cauchy Loss and Nonlocal Low-Rank Regularization

This work presents a novel robust image compressive sensing reconstruction approach. In contrast to the existing work, we employ the truncated Cauchy loss function to measure the errors induced during the measurement, showing strong robustness to impulsive noise and outliers. To ensure high quality reconstructed images, we utilize a non-local low rank regularizer - with truncated Schatten-p norm being the surrogate function of rank - to capture the self-similar property inherent in most natural images. Considering the fact that the whole optimization model is neither convex nor smooth, to solve it effectively, we firstly use the half-quadratic strategy to transform the loss function into a quadratic objective by introducing some auxiliary variables, and then iteratively and alternatively optimize different groups of variables. Extensive experimental results demonstrate its effectiveness in terms of both quantitative indexes of Peak Signal-to-Noise Ratio (PSNR) and Structural SIMilarity (SSIM), and visual quality under impulsive noise.