Schatten P-norm Minimization Research Articles

De Novo Prediction of Drug-Target Interactions Using Laplacian Regularized Schatten p-Norm Minimization.

In pharmaceutical sciences, a crucial step of the drug discovery is the identification of drug-target interactions (DTIs). However, only a small portion of the DTIs have been experimentally validated. Moreover, it is an extremely laborious, expensive, and time-consuming procedure to capture new interactions between drugs and targets through traditional biochemical experiments. Therefore, designing computational methods for predicting potential interactions to guide the experimental verification is of practical significance, especially for de novo situation. In this article, we propose a new algorithm, namely Laplacian regularized Schatten p-norm minimization (LRSpNM), to predict potential target proteins for novel drugs and potential drugs for new targets where there are no known interactions. Specifically, we first take advantage of the drug and target similarity information to dynamically prefill the partial unknown interactions. Then based on the assumption that the interaction matrix is low-rank, we use Schatten p-norm minimization model combined with Laplacian regularization terms to improve prediction performance in the new drug/target cases. Finally, we numerically solve the LRSpNM model by an efficient alternating direction method of multipliers algorithm. We evaluate LRSpNM on five data sets and an extensive set of numerical experiments show that LRSpNM achieves better and more robust performance than five state-of-the-art DTIs prediction algorithms. In addition, we conduct two case studies for new drug and new target prediction, which illustrates that LRSpNM can successfully predict most of the experimental validated DTIs.

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.

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.

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.

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.

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.

ELIMINACIÓN DEL RUIDO DE IMAGEN A TRAVÉS DE LA APROXIMACIÓN EXACTA DEL RANGO MÍNIMO CON REGULARIZACIÓN DE LA VARIACIÓN TOTAL RELATIVA

To Image denoising is one of the classical problems in image processing. Such denoising by minimum rank approximation via Schatten p-norm minimization is prone to cause over-smoothing. Intricate and irregular image structures cann't be distinguished dramatically by Schatten p-norm minimization. A flexible and precise model named weighted Schatten p-norm minimization (WSPM) with relative total variation regularization (RTV-WSPM) was proposed in this study to address this issue. The proposed RTV-WSPM not only had an accurate approximation with a Schatten p-norm but also considered the prior knowledge where different rank components have different importance by relative total variation. Moreover, the alternating direction method of multipliers was introduced to solve the proposed RTV-WSPM model. Experiments on Gaussian white noise and salt-and-pepper noise demonstrate that the proposed technique outperforms other state-of-the-art methods, especially under degradation for high-density image noise. In terms of peak signal-to-noise ratio evaluationthe proposed RTV-WSPM achieves significant improvements over the conventional WSPM under salt-and-pepper noise. Therefore, the RTV-WSPM exerts a good effect to restore the image structure and smoothness and improves denoising performances. Keywords: Weighted Schatten p-norm minimization; image denoising; low rank matrix approximation; RTV norm

Low-Rank Approximation via Generalized Reweighted Iterative Nuclear and Frobenius Norms.

The low-rank approximation problem has recently attracted wide concern due to its excellent performance in real-world applications such as image restoration, traffic monitoring, and face recognition. Compared with the classic nuclear norm, the Schatten-p norm is stated to be a closer approximation to restrain the singular values for practical applications in the real world. However, Schatten-p norm minimization is a challenging non-convex, non-smooth, and non-Lipschitz problem. In this paper, inspired by the reweighted ℓ1 and ℓ2 norm for compressive sensing, the generalized iterative reweighted nuclear norm (GIRNN) and the generalized iterative reweighted Frobenius norm (GIRFN) algorithms are proposed to approximate Schatten-p norm minimization. By involving the proposed algorithms, the problem becomes more tractable and the closed solutions are derived from the iteratively reweighted subproblems. In addition, we prove that both proposed algorithms converge at a linear rate to a bounded optimum. Numerical experiments for the practical matrix completion (MC), robust principal component analysis (RPCA), and image decomposition problems are illustrated to validate the superior performance of both algorithms over some common state-of-the-art methods.

Infrared small target detection via spatial–temporal infrared patch-tensor model and weighted Schatten p-norm minimization

Infrared small and dim target detection has been a typical application of infrared imaging techniques, and it still remains a challenging problem to achieve robust performance and fast processing capability under complex clutters and heavy noise. To cope with these obstacles, a novel detection method is proposed in this paper. Firstly, we construct a novel spatial–temporal infrared patch-tensor (STIPT) structure for mining valuable information in the time domain, and the small target can be detected after recovering and segmenting the low-rank background tensor. Then, considering the overshrinkage problem in low-rank component estimation field caused by vanilla nuclear norm and weighted nuclear norm, weighted Schatten p-norm is incorporated to improve the performance by considering the particular meaning of different singular values. A solution framework is proposed via Alternating Direction Method of Multipliers (ADMM), then an adaptive threshold is utilized to separate the targets. We take systematic analysis on real infrared data by the qualitative method and quantitative method, and the effectiveness and robustness of this method are verified in various scenarios.

Discriminative low-rank representation with Schatten-p norm for image recognition

Low-rank representation (LRR) has attracted much attention recently due to its efficacy in a rich variety of real world applications. Recently, the non-convex regularization has become widely used in the rank minimization problem. In this paper, we propose a discriminative low-rank representation with Schatten-p norm (DLRR-SPN) to learn a robust and discriminative affinity matrix for image recognition. To this end, we first impose the Schatten-p norm regularization on the representation matrix to learn the global structure of data. Moreover, the adaptive distance penalty is used to preserve the local neighbor relationship of data. The objective function is formulated as a Schatten-p norm minimization problem, which can be solved via alternating direction method of multipliers (ADMM). To enhance the separation ability of the discriminative affinity matrix for semi-supervised recognition problem, the angular information of the principal directions of the low-rank representation is further exploited. Finally, an effective semi-supervised classifier is utilized on the learned affinity matrix for final prediction. Extensive experimental results on image recognition demonstrate the effectiveness of the proposed method and its superiority in performance over the related state-of-the-art methods.

Image denoising in impulsive noise via weighted Schatten p-norm regularization

Low-rank methods have been widely exploited in image denoising and have shown admirable denoising performance, of which weighted Schatten p-norm minimization (WSNM) is particularly effective. However, the WSNM method which applies Frobenius-norm loss model cannot obtain a satisfactory denoising performance when images corrupted by impulse noise. An optimization strategy based on the alternating direction method of multipliers framework is used to solve the proposed model efficiently. Experimental results show that the proposed method outperforms some state-of-the-art denoising methods both quantitatively and qualitatively under various impulsive noise models.

A new development of non-local image denoising using fixed-point iteration for non-convex ℓp sparse optimization.

We proposed a new efficient image denoising scheme, which mainly leads to four important contributions whose approaches are different from existing ones. The first is to show the equivalence between the group-based sparse representation and the Schatten-p norm minimization problem, so that the sparsity of the coefficients for each group can be measured by estimating the underlying singular values. The second is that we construct the proximal operator for sparse optimization in ℓp space with p ∈ (0, 1] by using fixed-point iteration and obtained a new solution of Schatten-p norm minimization problem, which is more rigorous and accurate than current available results. The third is that we analyze the suitable setting of power p for each noise level σ = 20, 30, 50, 60, 75, 100, respectively. We find that the optimal value of p is inversely proportional to the noise level except for high level of noise, where the best values of p are 1 and 0.95, when the noise levels are respectively 75 and 100. Last we measure the structural similarity between two image patches and extends previous deterministic annealing-based solution to sparsity optimization problem through incorporating the idea of dictionary learning. Experimental results demonstrate that for every given noise level, the proposed Spatially Adaptive Fixed Point Iteration (SAFPI) algorithm attains the best denoising performance on the value of Peak Signal-to-Noise Ratio (PSNR) and structure similarity (SSIM), being able to retain the image structure information, which outperforms many state-of-the-art denoising methods such as Block-matching and 3D filtering (BM3D), Weighted Nuclear Norm Minimization (WNNM) and Weighted Schatten p-Norm Minimization (WSNM).

Weighted Schatten p-norm minimization for 3D magnetic resonance images denoising

Magnetic resonance (MR) imaging plays an important role in clinical diagnosis and scientific research. A clean MR image can better provide patient's information to doctors or researchers for further treatment. However, in real life, MR images are inevitably corrupted by annoying Rician noise in the process of imaging. Aiming at the Rician noise of 3D MR images, a framework is proposed to suppress noise by low-rank matrix approximation (LRMA) with weighted Schatten p-norm minimization regularization (WSNMD-3D). The proposed method not only considers the importance of different rank components, but can also approximate the true rank of the latent low-rank matrix. This approach first groups similar non-local cubic patches extracted from the noisy 3D MR image into a matrix whose columns are vectorized patches. The above matrix can be modeled as a low-rank matrix approximate model. Then weighted Schatten p-norm minimization (WSNM) is applied to the model, which shrinks different rank components with different treatments. Finally, the denoised 3D MR image is acquired by aggregating all denoised patches with weighted averaging. Experimental results on synthetic and real 3D MR data show that the proposed method obtains better results than state-of-the-art methods, both visually and quantitatively.

Image compressed sensing based on non-convex low-rank approximation

Nonlocal sparsity and structured sparsity have been evidenced to improve the reconstruction of image details in various compressed sensing (CS) studies. The nonlocal processing is achieved by grouping similar patches of the image into the groups. To exploit these nonlocal self-similarities in natural images, a non-convex low-rank approximation is proposed to regularize the CS recovery in this paper. The nuclear norm minimization, as a convex relaxation of rank function minimization, ignores the prior knowledge of the matrix singular values. This greatly restricts its capability and flexibility in dealing with many practical problems. In order to make a better approximation of the rank function, the non-convex low-rank regularization namely weighted Schatten p-norm minimization (WSNM) is proposed. In this way, both the local sparsity and nonlocal sparsity are integrated into a recovery framework. The experimental results show that our method outperforms the state-of-the-art CS recovery algorithms not only in PSNR index, but also in local structure preservation.

Matrix Completion with Ordering Relation Constraints and its Applications

We relax the equality constraints in the very general and well-known affine Schatten p-norm minimization problem into complete loss function-based constraints. Owing to the imposed equality constraints, existing methods only have limited degree of model flexibility, via their optimization of the objective energy function. By our proposed transformation, the decision variables in the objective function can directly achieve L0 norm minimization via the process of enumerating the matrix rank (i.e. matrix ordering constraint). We show that, our new objective function is still reasonable, and its minimum can be obtained by a more general form of the Fixed-Point Continuation framework with almost the same computational cost at each matrix order enumeration. Experiments show that, our algorithm has good performance compared to its predecessor over some datasets and applications.

New RIC Bounds for Low Rank Matrix Recovery via Schatten-p Minimization with 0 < p < 1

Low rank matrix recovery is widely applicable in many fields and has been the focus of much research recently. Schatten-p (0 < p ≤ 1) norm minimization are two main recovery methods. For case p = 1, i.e. nuclear norm minimization, there have been abundant results on exact and stable recovery conditions. The restricted isometry constants (RIC) play an important role in exact recovery theory via nuclear norm minimization. However, there are few results on Schatten-p (0 < p < 1) norm minimization. In this paper, we establish new RIC bounds to guarantee exact recovering low rank matrices via Schatten-p (0 < p < 1) norm minimization. First, we provide the RIC bound of δ[g(p,2r)](t-1) + 2r to guarantee exact recovery. Especially, the exact recovery can be succeed via Schatten-1/2 minimization if for any t > 1. Second, we obtain the RIC bound of δ2r to guarantee exact recovery. For Schatten-1/2 minimization, the bound is δ2r < 1/3. Moreover, we get the stable recovery results with bounded noise.

Exact minimum rank approximation via Schatten [formula omitted]-norm minimization

Minimizing the rank of a matrix with a given system of affine constraints is to recover the lowest-rank matrix with many important applications in engineering and science. A convex relaxation of the rank minimization problem by minimizing the nuclear norm instead of the rank of the matrix has recently been proposed. Recht and Fazel concluded that nuclear norm minimization subject to affine constraints is equivalent to rank minimization under a certain condition in terms of the rank-restricted isometry property. In this paper, we extend some recent results from nuclear norm minimization to Schatten p-norm minimization and present a theoretical guarantee for the Schatten p-norm minimization if a certain restricted isometry property holds for the linear affine transform. Our results improve on the previous works where recovery is used for nuclear norm minimization. An algorithm based on the Majorization Minimization algorithm has been proposed to solve Schatten p-norm minimization. By using an approximate singular value decomposition procedure, we obtain a fast and robust algorithm. The numerical results on the matrix completion problem with noisy measurements indicate that our algorithm gives a more accurate reconstruction and takes less time compared with some other existing algorithms.

EXACT LOW-RANK MATRIX RECOVERY VIA NONCONVEX SCHATTEN p-MINIMIZATION

The low-rank matrix recovery (LMR) arises in many fields such as signal and image processing, quantum state tomography, magnetic resonance imaging, system identification and control, and it is generally NP-hard. Recently, Majumdar and Ward [Majumdar, A and RK Ward (2011). An algorithm for sparse MRI reconstruction by Schatten p-norm minimization. Magnetic Resonance Imaging, 29, 408–417]. had successfully applied nonconvex Schatten p-minimization relaxation of LMR in magnetic resonance imaging. In this paper, our main aim is to establish RIP theoretical result for exact LMR via nonconvex Schatten p-minimization. Carefully speaking, letting [Formula: see text] be a linear transformation from ℝm×n into ℝs and r be the rank of recovered matrix X ∈ ℝm×n, and if [Formula: see text] satisfies the RIP condition [Formula: see text] for a given positive integer k ∈ {1, 2, …, m – r}, then r-rank matrix can be exactly recovered. In particular, we obtain a uniform bound on restricted isometry constant [Formula: see text] for any p ∈ (0, 1] for LMR via Schatten p-minimization.

FOCUSS Based Schatten-p Norm Minimization for Real-Time Reconstruction of Dynamic Contrast Enhanced MRI

This work addresses the problem of real-time reconstruction of Dynamic Contrast Enhanced Magnetic Resonance Images (DCE MRI) from partially sampled K-space. The difference image between the current and previous frame is modeled as a rank deficient matrix. It is reconstructed from via nonconvex Schatten-p norm minimization. We develop a FOCUSS (FOCally Under-determined System Solver) based algorithm for solving the said minimization problem. The proposed technique has been compared with state-of-the-art online and offline reconstruction techniques. It yields slightly worse reconstruction results than the offline technique but considerably superior results, both in terms of accuracy and speed compared to the online technique.

An algorithm for sparse MRI reconstruction by Schatten p-norm minimization

In recent years, there has been a concerted effort to reduce the MR scan time. Signal processing research aims at reducing the scan time by acquiring less K-space data. The image is reconstructed from the subsampled K-space data by employing compressed sensing (CS)-based reconstruction techniques. In this article, we propose an alternative approach to CS-based reconstruction. The proposed approach exploits the rank deficiency of the MR images to reconstruct the image. This requires minimizing the rank of the image matrix subject to data constraints, which is unfortunately a nondeterministic polynomial time (NP) hard problem. Therefore we propose to replace the NP hard rank minimization problem by its nonconvex surrogate — Schatten p-norm minimization. The same approach can be used for denoising MR images as well.Since there is no algorithm to solve the Schatten p-norm minimization problem, we derive an efficient first-order algorithm. Experiments on MR brain scans show that the reconstruction and denoising accuracy from our method is at par with that of CS-based methods. Our proposed method is considerably faster than CS-based methods.