Schatten P-norm Research Articles

Tensor multi-view clustering method for natural image segmentation

Image segmentation methods based on clustering have become a advancement in the field of computer vision, and the performance of these methods mainly depends on the performance of the clustering algorithm. However, these methods have some shortcomings. First, a single feature cannot accurately obtain the complete information of the superpixels. Second, the commonly used Euclidean distance cannot extract the nonlinear manifold structure between superpixels. Third, the high-dimensional information between superpixels is not considered. Therefore, this article proposes a Tensor Multi-view Clustering Method for Natural Image Segmentation (TMCNIS), in which firstly the multiclass features of the superpixels are extracted to obtain complete information. Secondly, the Jensen–Shannon divergence is used to delineate the relationship between superpixels to obtain a more complete nonlinear structure. Thirdly, an adaptively weighted tensor Schatten-p norm is proposed to better approximate the target rank of the tensor. It also automatically shrinks the singular values appropriately to construct a finer tensor, thus fully capturing the spatial structure in the tensor. Experimental results on four clustering datasets and two image segmentation datasets demonstrate the superiority of TMCNIS compared to existing methods.

Incomplete multi-view clustering via confidence graph completion based tensor decomposition

In recent years, extensive incomplete multi-view clustering models have been proposed to solve the problem of real-world multi-view data with missing views. However, they still have the following two drawbacks: (1) They ignore the harmful effects of outliers in data and the noise generated during graph recovery. (2) They do not fully explore the high-order relationships among views. To this end, we design a novel graph learning model called confidence graph completion based tensor decomposition (CGCTD) for incomplete multi-view clustering. Specifically, we use the confidence graphs to guide the learning of the complete graphs, which reduces the detrimental effects of outliers and missing samples in the data. Then, we stack the complete graphs of each view into an original tensor to explore high-order relationships and correlations between views. To reduce the negative effects of noise, we decompose the original tensor into an essential tensor and a noise tensor. The essential tensor is introduced to recover the accurate affinity graphs, and the noise tensor aims to model the noise contained in the corrupted graphs. Furthermore, we impose the tensor Schatten p-norm constraint on the essential tensor, which can enhance the low-rank property of the graphs and explore the similarity structure between views. Through extensive experiments on eight benchmark datasets, we demonstrate that the proposed CGCTD outperforms several existing state-of-the-art methods.

Dual-dual subspace learning with low-rank consideration for feature selection

The performance of machine learning algorithms can be affected by redundant features of high-dimensional data. Furthermore, these irrelevant features increase the time of computation for learning model. These problems can be addressed by leveraging different techniques including feature selection and dimensionality reduction. Unsupervised feature selection has drawn increased attention due to the difficulty of label collection for supervised feature selection. To this end, we developed an innovative approach based on nonnegative matrix factorization (NMF) to remove redundant information. In this technique, for the first time, the local information preserving regularization and global information preserving regularization are applied for both feature weight matrix and representation matrix which is why we called Dual-Dual regularized feature selection. Furthermore, Schatten p-norm is utilized to extract inherent low-rank properties of data. To demonstrate the effectiveness of the proposed method, experimental studies are conducted on six benchmark datasets. The computational results show that the proposed method in comparison with state-of-the-art unsupervised feature selection techniques is more efficient for feature selection.

Multiview Subspace Clustering via Low-Rank Symmetric Affinity Graph.

Multiview subspace clustering (MVSC) has been used to explore the internal structure of multiview datasets by revealing unique information from different views. Most existing methods ignore the consistent information and angular information of different views. In this article, we propose a novel MVSC via low-rank symmetric affinity graph (LSGMC) to tackle these problems. Specifically, considering the consistent information, we pursue a consistent low-rank structure across views by decomposing the coefficient matrix into three factors. Then, the symmetry constraint is utilized to guarantee weight consistency for each pair of data samples. In addition, considering the angular information, we utilize the fusion mechanism to capture the inherent structure of data. Furthermore, to alleviate the effect brought by the noise and the high redundant data, the Schatten p-norm is employed to obtain a low-rank coefficient matrix. Finally, an adaptive information reduction strategy is designed to generate a high-quality similarity matrix for spectral clustering. Experimental results on 11 datasets demonstrate the superiority of LSGMC in clustering performance compared with ten state-of-the-art multiview clustering methods.

Low-rank tensor completion via nonlocal self-similarity regularization and orthogonal transformed tensor Schatten-p norm

Low-rank tensor completion via nonlocal self-similarity regularization and orthogonal transformed tensor Schatten-p norm

Tensor nonconvex unified prior for tensor recovery

Tensor data, such as hyperspectral images and multi-frame videos, have gained significant attention in practical applications. However, the inherent degradation phenomena during data acquisition, including noise and missing pixels, give rise to a series of ill-posed inverse problems that need to be addressed. Currently, the rational exploration of prior knowledge for tensor recovery, including global low-rankness and local smoothness, has emerged as a common concern. Inspired by recent notable works, this paper proposes a novel tensor non-convex unified prior term, which employs weighted tensor Schatten p-norm as a rank surrogate function in the gradient domain. The new prior can yield a regularizer that effectively captures low-rankness and smoothness, and is applied to tensor completion and tensor robust principal component analysis models. An efficient algorithm is developed by using the alternating direction method of multipliers and its convergence analysis is also provided. Extensive experimental results demonstrate that the proposed method outperforms the state-of-the-art methods, particularly in cases of high missing rates and strong noise levels.

Multi-view reduced dimensionality K-means clustering with [formula omitted]-norm and Schatten [formula omitted]-norm

Recently, multi-view high dimensional data obtained from diverse domains or various feature extractors has drawn great attention due to its reflection of different properties or distributions. In this paper, we propose a novel unsupervised multi-view clustering method, which is called Multi-View Reduced Dimensionality K-means clustering (MRDKM) and integrates the dimension reduction mechanism, σ-norm, Schatten p-norm, and multi-view K-means clustering. Moreover, an unsupervised optimization scheme was proposed to solve the minimization problem with good convergence properties. Comprehensive evaluations of five benchmark datasets and comparisons with several multi-view clustering algorithms demonstrate the superiority of the proposed work.

EMCMDA: predicting miRNA-disease associations via efficient matrix completion

Abundant researches have consistently illustrated the crucial role of microRNAs (miRNAs) in a wide array of essential biological processes. Furthermore, miRNAs have been validated as promising therapeutic targets for addressing complex diseases. Given the costly and time-consuming nature of traditional biological experimental validation methods, it is imperative to develop computational methods. In the work, we developed a novel approach named efficient matrix completion (EMCMDA) for predicting miRNA-disease associations. First, we calculated the similarities across multiple sources for miRNA/disease pairs and combined this information to create a holistic miRNA/disease similarity measure. Second, we utilized this biological information to create a heterogeneous network and established a target matrix derived from this network. Lastly, we framed the miRNA-disease association prediction issue as a low-rank matrix-complete issue that was addressed via minimizing matrix truncated schatten p-norm. Notably, we improved the conventional singular value contraction algorithm through using a weighted singular value contraction technique. This technique dynamically adjusts the degree of contraction based on the significance of each singular value, ensuring that the physical meaning of these singular values is fully considered. We evaluated the performance of EMCMDA by applying two distinct cross-validation experiments on two diverse databases, and the outcomes were statistically significant. In addition, we executed comprehensive case studies on two prevalent human diseases, namely lung cancer and breast cancer. Following prediction and multiple validations, it was evident that EMCMDA proficiently forecasts previously undisclosed disease-related miRNAs. These results underscore the robustness and efficacy of EMCMDA in miRNA-disease association prediction.

Nonconvex submodule clustering via joint sliced sparse gradient and cluster-aware approach

Most existing subspace clustering methods preprocess image data by converting them into vectors, which lacks exploration of the spatial structure of high-dimensional data. Therefore, we proposes a nonconvex submodule clustering model (NSSGCA) via joint sliced sparse gradient and cluster-aware approach. NSSGCA arranges each 2D image as lateral slices of a 3rd-order tensor, and utilizes the t-product under the model of the union of free submodules to represent 3rd-order tensor samples, thereby exploring the latent spatial structure of samples. To more accurately approximate tensor rank, a nonconvex Schatten p-norm constraint is imposed on the rotated representation tensor. Under the submodule framework, a consistent gradient matrix is derived based on the δ-nearest neighbor adjacency graph to construct sliced sparse gradient (SSG) regularization, which is more conducive to clustering tasks. NSSGCA learns representation tensor with clearer block-like structure based on ℓq norm and cluster-aware attention mechanism. The convergence of the constructed sequence to the stationary Karush–Kuhn–Tucker (KKT) point is proven. Experimental results on real-world image datasets confirm the effectiveness of NSSGCA.

The Application of the Accelerated Proximal Gradient Descent Algorithm for the Solution of the Weighted Schatten-p Norm in Sparse Noise Extraction

The Application of the Accelerated Proximal Gradient Descent Algorithm for the Solution of the Weighted Schatten-p Norm in Sparse Noise Extraction

Comprehensive multi-view self-representations for clustering

Subspace learning-based methods have shown excellent performance for multi-view clustering, yet have the following problems: (1) most existing methods obtain the subspace representation from the original space, which might contain noises and cannot guarantee a clean enough subspace representation; (2) existing methods mainly focus on the consistency of the subspace representation, while the unique information of each view is not sufficiently exploited. To solve these two problems, we propose a novel multi-view subspace clustering method called comprehensive multi-view self-representations (CMSR). Specifically, we learn the original coefficient matrix of each view through the self-representation, which can reduce the noise of the original space to some extent. Then, we learn the subspace representation of the original coefficient matrix and decompose it into a consistent coefficient matrix and multiple diverse coefficient matrices, which can exploit the consistent and complementary information of multi-view data. Further, we impose the Schatten p-norm constraint on the consistent coefficient matrix to capture robust consistent information. Finally, the comprehensive results on eight real datasets demonstrate the versatility and effectiveness of the proposed method.

Tensor schatten-p norm guided incomplete multi-view self-representation clustering

Incomplete multi-view clustering (IMC), which aims to handle incomplete data with missing views, has recently received more attention. According to the investigation of the existing IMC methods, they have at least one of the four limitations. First, some methods do not recover the missing view-specific information. Second, some approaches rely on fixed similarity graphs of incomplete views. Third, some methods need a separate spectral clustering step to produce the final cluster indicator. Last, most works utilize tensor nuclear norm to characterize the low-rank property of multi-view representation. These four limitations degrade the clustering performance. Hence, this paper proposes a tensor schatten-p norm guided incomplete multi-view self-representation clustering (TIMSRC) method to handle these four limitations simultaneously. TIMSRC models the missing instances as a data matrix for each view. And it integrates the missing samples inferring and complete self-representation graph learning into a unified framework. TIMSRC adopts a multi-view spectral clustering with adaptive weights to mine the semantic information while considering the contributions of multiple self-representing graphs. TIMSRC exploits the tensor schatten-p norm to enhance the low-rank property of a three-order tensor constructed with multiple self-representation graphs. We present an optimization algorithm to solve the objective function. The experiments demonstrate that TIMSRC outperforms the state-of-the-art IMC approaches.

Spinet-QSM: model-based deep learning with schatten p-norm regularization for improved quantitative susceptibility mapping.

Quantitative susceptibility mapping (QSM) provides an estimate of the magnetic susceptibility of tissue using magnetic resonance (MR) phase measurements. The tissue magnetic susceptibility (source) from the measured magnetic field distribution/local tissue field (effect) inherent in the MR phase images is estimated by numerically solving the inverse source-effect problem. This study aims to develop an effective model-based deep-learning framework to solve the inverse problem of QSM. This work proposes a Schatten -norm-driven model-based deep learning framework for QSM with a learnable norm parameter to adapt to the data. In contrast to other model-based architectures that enforce the l -norm or l -norm for the denoiser, the proposed approach can enforce any -norm ( ) on a trainable regulariser. The proposed method was compared with deep learning-based approaches, such as QSMnet, and model-based deep learning approaches, such as learned proximal convolutional neural network (LPCNN). Reconstructions performed using 77 imaging volumes with different acquisition protocols and clinical conditions, such as hemorrhage and multiple sclerosis, showed that the proposed approach outperformed existing state-of-the-art methods by a significant margin in terms of quantitative merits. The proposed SpiNet-QSM showed a consistent improvement of at least 5% in terms of the high-frequency error norm (HFEN) and normalized root mean squared error (NRMSE) over other QSM reconstruction methods with limited training data.

Orthogonal subspace exploration for matrix completion

Matrix completion, aiming at restoring a low-rank matrix from observed entries, indicates the connection with the subspace clustering due to the low-rank property. However, it is expensive to incorporate subspace learning into the pervasive surrogate of matrix completion, the nuclear norm. In this paper, we design an orthogonal subspace exploration model for matrix completion, which can be easily integrated due to the succinct formulation. Then, we propose a non-convex surrogate with tractable solutions for low-rank matrix completion, so that the subspace exploration can be performed simultaneously. Compared with the existing surrogates (e.g., nuclear norm, Schatten-p norm, max norm, etc.), the proposed surrogate is differential such that the optimization is still simple even after the subspace exploration is incorporated. Although the surrogate is non-convex, a parameter-free algorithm that is proved to converge into the global optimum is developed. The optimization consists of closed-form solutions so that the orthogonal subspace exploration will not distinctly bring additional costs and the algorithm empirically converges within dozens of iterations. Experiments illustrate the efficiency and superiority of our model.

A Bisection method for computing the proximal operator of the [formula omitted]-norm for any [formula omitted] with application to Schatten [formula omitted]-norms

The computation of the proximal operator of the ℓp-norm (0<p<1) is critically important for non-convex optimization with the sparse-promoting ℓp-regularization. Over the last decade, many exact and inexact numerical methods for simple and efficient computation of the proximal operator have been proposed in the literature. For this purpose, a Bisection method is given in the paper, which is based on the properties of the proximal operator. Numerical experiments illuminate the efficiency of the Bisection method. As a byproduct, the Bisection method is applied to calculate the proximal operator of the Schatten p-norm with 0<p<1.

Embedded Feature Selection on Graph-Based Multi-View Clustering

Recently, anchor graph-based multi-view clustering has been proven to be highly efficient for large-scale data processing. However, most existing anchor graph-based clustering methods necessitate post-processing to obtain clustering labels and are unable to effectively utilize the information within anchor graphs. To solve these problems, we propose an Embedded Feature Selection on Graph-Based Multi-View Clustering (EFSGMC) approach to improve the clustering performance. Our method decomposes anchor graphs, taking advantage of memory efficiency, to obtain clustering labels in a single step without the need for post-processing. Furthermore, we introduce the l2,p-norm for graph-based feature selection, which selects the most relevant data for efficient graph factorization. Lastly, we employ the tensor Schatten p-norm as a tensor rank approximation function to capture the complementary information between different views, ensuring similarity between cluster assignment matrices. Experimental results on five real-world datasets demonstrate that our proposed method outperforms state-of-the-art approaches.

Anchor graph-based multiview spectral clustering

Significant advances in graph-oriented clustering methods can be attributed to their effectiveness in leveraging relationships and complex structures within multiview data. However, several limitations persist in most existing graph-based multiview clustering approaches. First, quadratic or cubic complexity is required for graph construction or eigendecomposition of the Laplacian matrix in many existing methods. Second, certain methods overlook the differences between views and employ an identical indicator matrix, which can lead to over-learning in practical scenarios. Third, existing methods often neglect spatial structure and complementary information, focusing primarily on calculating error feature-by-feature using different norms. In order to tackle these drawbacks, we propose a new multiview spectral clustering model called Anchor Graph-based Multiview Spectral Clustering(AG-MSC). AG-MSC incorporates an adaptive weighting mechanism that assigns weights to each view, enhancing the robustness of the algorithm. Using a tensor Schatten p-norm constraint minimizes the discrepancy between indicator matrices obtained from different views, thereby preserving high-order information and spatial structure. To improve computational efficiency, we replace the full adjacency matrices of the corresponding views with anchor graphs. AG-MSC offers a distinct advantage over conventional spectral clustering by directly obtaining all sample categories without additional post-processing steps. We have validated the efficiency of our method through extensive experimental evaluations.

Tensorized Label Learning on Anchor Graph

Graph-based multimedia data clustering has attracted much attention due to the impressive clustering performance for arbitrarily shaped multimedia data. However, existing graph-based clustering methods need post-processing to get labels for multimedia data with high computational complexity. Moreover, it is sub-optimal for label learning due to the fact that they exploit the complementary information embedded in data with different types pixel by pixel. To handle these problems, we present a novel label learning model with good interpretability for clustering. To be specific, our model decomposes anchor graph into the products of two matrices with orthogonal non-negative constraint to directly get soft label without any post-processing, which remarkably reduces the computational complexity. To well exploit the complementary information embedded in multimedia data, we introduce tensor Schatten p-norm regularization on the label tensor which is composed of soft labels of multimedia data. The solution can be obtained by iteratively optimizing four decoupled sub-problems, which can be solved more efficiently with good convergence. Experimental results on various datasets demonstrate the efficiency of our model.

Multi-view Subspace Clustering Based on Weighted Tensor Schatten-P Norm

In the era of advancing technology and continuous societal evolution, an abundance of voluminous data from various domains has become prevalent in the public domain. Managing this wealth of data necessitates a series of preprocessing procedures, encompassing tasks such as data storage, computation, and analysis, with the aim of unveiling intrinsic features and patterns contained within. Among these procedures, clustering stands out as a crucial step, with the primary goal of unveiling the hidden structures within the data and organizing data samples into meaningful clusters based on specific criteria. Early clustering efforts were based on a singleview. Real world data, however, often contains interrelated information, leading to the emergence of multi-view clustering. Multiview subspace clustering (T-SVD) has been widely used in recent years. However, the equal treatment of each singular value in this approach lacks relevance when the data is noisy or contaminated. In order to improve the robustness and clustering performance, this paper presents a T-SVD weighted tensor norm. In order to improve the robustness, a Schatten-P norm was proposed. This paper explores multi-view subspace clustering is based on the Schatten-P norm weighted tensor. Experiments are carried out on 6 benchmark data sets, and the results are compared with 4 existing ones. Experiments show that the proposed approach is better than the most advanced multi-view subspace clustering methods.

Quaternion weighted Schatten [formula omitted]-norm minimization for color image restoration with convergence guarantee

Over the past decades, the rank approximation issue have been extensively investigated, among which weighted nuclear norm minimization (WNNM) and weighted Schatten p-norm minimization (WSNM) are two prevailing methods and have shown great superiority in various image restoration (IR) problems. However, for the complicated color image restoration (CIR) problems, traditional WNNM/WSNM method only processes three color channels individually and fails to consider their cross-channel correlations. Very recently, a quaternion-based WNNM approach (QWNNM) has been developed to mitigate this issue, which is capable of representing the color image as a whole in the quaternion domain. Despite QWNNM’s empirical success, its convergence behavior has not been rigorously studied. The main contributions of this paper are twofold. Firstly, we extend the WSNM into quaternion domain and correspondingly propose a novel quaternion-based WSNM model (QWSNM) for tackling the CIR problems. Extensive experiments on two representative CIR tasks, including color image denoising and deblurring, demonstrate that the proposed QWSNM method performs favorably against many state-of-the-art alternatives, in both quantitative and qualitative evaluations. Secondly, we provide a preliminary theoretical convergence analysis. By modifying the quaternion alternating direction method of multipliers (QADMM) through a simple continuation strategy, we theoretically prove the fixed-point convergence property of the iterative sequences generated by QWNNM and QWSNM. The source code of our algorithm can be available at the website: https://github.com/qiuxuanzhizi/QWSNM.