Matrix Decomposition Problem Research Articles

Secure Key Exchange in Tropical Cryptography: Leveraging Efficiency with Advanced Block Matrix Protocols

In the quest for robust and efficient digital communication, this paper introduces cutting-edge key exchange protocols leveraging the computational prowess of tropical semirings and the structural resilience of block matrices. Moving away from the conventional use of finite fields, these protocols deliver markedly faster processing speeds and heightened security. We present two implementations of our concept, each utilizing a different platform for the set of commuting matrices: one employing tropical polynomials of matrices and the other employing Linde–de la Puente matrices. The inherent simplicity of tropical semirings leads to a decrease in operational complexity, while using block matrices enhances our protocols’ security profile. The security of these protocols relies on the Matrix Decomposition Problem. In addition, we provide a comparative analysis of our protocols against existing matrix block-based protocols in finite fields. This research marks a significant shift in cryptographic protocol design, is specifically tailored for demanding engineering applications, and sets a new standard in secure and efficient digital communication.

A simple method to construct multivariate dual framelets with high-order vanishing moments

When constructing multivariate framelets, it is often unavoidable to work with matrices of multivariate trigonometric polynomials and complicated matrix decomposition problems. These problems become even harder when good properties such as high-order vanishing moments are required on the framelets. In this paper, we establish a new method for constructing multivariate dual framelets with high-order vanishing moments. The underlying scheme of our algorithm is the famous Mixed Extension Principle that allows us to derive the high-pass filters (or framelet generators) from a given pair of refinable filters with high-order linear-phase moments. Our method only involves two steps: (1) directly constructing the first few pairs of high-pass filters by using the linear-phase moment conditions of the refinement filters; (2) solving a system of linear equations to obtain the rest of the high-pass filters. Both are easy to implement for scientific computation, regardless of what dimension or dilation matrix we work with. Apart from high-order vanishing moments, we will see that if the refinement filters take coefficients from some subfield [Formula: see text] of [Formula: see text] that is closed under complex conjugation, so do the high-pass filters. Furthermore, our algorithm gives the upper bounds for the number of high-pass filters in arbitrary dimensions. At the end of the paper, we will give several illustrative examples, from which we can also see that the support sizes of the high-pass filters are comparable with those of the refinement filters.

Asymmetric Cryptography Based on the Tropical Jones Matrix

In recent years, the tropical polynomial factorization problem, the tropical matrix decomposition problem, and the tropical multivariate quadratic equation solving problem have been proved to be NP-hard. Some asymmetric cryptographic systems based on tropical semirings have been proposed, but most of them are insecure and have been successfully attacked. In this paper, a new key exchange protocol and a new encryption protocol are proposed based on the difficulty of finding the multiple exponentiation problem of the tropical Jones matrices. The analysis results indicate that our protocol can resist various existing attacks. The complexity of attacking an MEP by adversaries is raised due to the larger number of combinations in the tropical Jones matrices compared to regular matrix polynomials. Furthermore, the index semiring is the non-negative integer cyclic matrix semiring, leading to a higher efficiency in key generation.

A Study of Marketing Capabilities and Their Performance Impact Based on SWOT Matrix Modeling

Abstract Marketing capability is a vital resource capability, this paper first utilizes the gradient descent algorithm to calculate the matrix decomposition problem and mine the relevant data of marketing capability. Then combined with the ANP algorithm, it solves the problem of the SWOT algorithm being unquantifiable. It constructs a SWOT-ANP strategy analysis model, and empirically examines the influence of enterprise strategy and performance. The model is applied to the enterprise’s marketing capability, with performance relationship as the target layer, SWOT as the criterion layer, and the weight values of 15 influencing factors are calculated through the network hierarchy. The weight of the opportunity layer in the local weight is higher than that of the disadvantage layer, which leads to the influence of sub-influence factors, and the highest weight of the disadvantage influence factor is 0.04388 lower than the lowest influence factor of opportunity. Combining with the factor intensity score value, the integrated performance influence of O, T, S, and W is calculated to be 8.438, 6.00747, 7.42637, and 7.69062, respectively, and the sample is obtained through the strategy positioning The development strategy of the enterprise is S-T strategy.

Performance evaluation of LU matrix decomposition using the SYCL standard

The object of this study is the performance of the SYCL standard tools when solving the LU matrix decomposition problem. SYCL is a fairly new technology for parallel computing in heterogeneous systems, so the topic of evaluating the performance of the standard on specific tasks in the field of parallel computing is relevant. In the study, the algorithm of parallelized LU decomposition of a square matrix was implemented by means of the SYCL standard and standard C++, and an experiment was conducted to test the implementation in a heterogeneous system with several types of processors. During testing, the program received square matrices of various dimensions as input, and the output was the execution time of the LU schedule on the selected processor. The obtained results, presented in the form of tabular and graphic data, show the advantage of the implementation of the SYCL standard over ordinary C++ by more than 2 times when using a graphics processor. It was experimentally shown that the implementation on SYCL is almost not inferior in speed to the implementation on ordinary C++ when executed on a central processor. Such results are caused both by the high possibility of parallelizing the LU schedule algorithm itself, and by the great work of the developers of the standard on its optimization. The obtained results indicate the possibility of speeding up the solution of the LU decomposition of the matrix and similar algorithms by means of SYCL when using heterogeneous systems with processors optimized for data parallelism. The results of the study can be used in justifying the choice of technology for solving LU matrix decomposition problems or problems with a similar parallelization scheme.

Farthest-Nearest Distance Neighborhood and Locality Projections Integrated With Bootstrap for Industrial Process Fault Diagnosis

It has become a big challenge and a hot topic of research to capture the most relevant features from high-dimensional process data for enhancing fault diagnosis. To effectively extract discriminative features from high-dimensional data, a novel dimensionality reduction (DR) approach named neighborhood and locality projections with the farthest and nearest distance (FNDNLP) is first proposed for industrial process fault feature acquisition and diagnosis. By constructing intra-class weights and inter-class weights, FNDNLP takes both the intra-class distance and the inter-class distance into consideration in its objective function, improving the diagnostic ability of extracted features through maximizing the inter-class distance and minimizing the intra-class distance. In addition, bootstrap based FNDNLP (BFNDNLP) is further proposed to handle the matrix decomposition problem in FNDNLP. To find the proper order through DR, the Akaike information criterion (AIC) is adopted. Finally, the Naïve Bayes (NB) based classifier is utilized to achieve acceptable fault diagnosis. The simulation results from two complex industrial cases indicate that the proposed methodology can achieve higher diagnosis accuracy than other related methods. What's more, the DR features are further analyzed to show the effectiveness and benefits of the proposed BFNDNLP extraction approach. The main source code of this paper can be obtained on GitHub soon.

Small Target Detection Method Based on Low-Rank Sparse Matrix Factorization for Side-Scan Sonar Images

Target detection in side-scan sonar images plays a significant role in ocean engineering. However, the target images are usually severely interfered by the complex background and strong environmental noise, which makes it difficult to extract robust features from small targets and makes the target detection task quite challenging. In this paper, a novel small target detection method in sonar images is proposed based on the low-rank sparse matrix factorization. Initially, the side-scan sonar images are preprocessed so as to highlight the individual differences of the target. Then, the problems of target feature extraction and noise removal are characterized as the problem of matrix decomposition. An improved Robust Principal Component Analysis algorithm is used to extract target information, and the fast proximal gradient method is used to optimize the solution. The original sonar image is reconstructed into the low-rank background matrix, the sparse target matrix, and the noise matrix. Eventually, a morphological operation is used to filter out the noise and refine the target edges in the target matrix for improving the accuracy of target detection. Experimental results show that the proposed method not only achieves better detection performance in comparison to the conventional baseline algorithms but also performs robustly in various signal-to-clutter ratio conditions.

Novel Discriminant Locality Preserving Projection Integrated With Monte Carlo Sampling for Fault Diagnosis

In complex industrial processes, the technique of fault diagnosis has been playing an increasingly considerable role in ensuring the safety of life and property. Unfortunately, the process data of complex industrial processes have the features of high dimension. Feature extraction from high-dimensional data is promising to coping with the fault data with high dimension. Recently, one of manifold learning methods named discriminant locality preserving projection achieves excellent performance in feature extraction. However, the performance of discriminant locality preserving projection (DLPP) is subject to the problem of matrix decomposition in the denominator of the objection function caused by the small sample size (SSS) issue. To overcome this limitation, novel DLPP integrated with Monte Carlo sampling is proposed to enhance the performance of feature extraction through dimensionality reduction. In the proposed MC-DLPP, Monte Carlo sampling is first utilized to generate fault samples for each fault type. With the aid of the virtually generated fault samples, the rank of the matrix in the denominator of the objection function of DLPP increases, thus well addressing the SSS problem. The Softmax classifier is used for fault diagnosis. To test the performance of the improved DLPP-based fault diagnosis, case studies using the Tennessee Eastman process are carried out. Simulation results confirm the presented MC-DLPP achieves superior accuracy in fault diagnosis.

SINGLE IMAGE SUPER RESOLUTION VIA COUPLED SPARSE AND LOW RANK DICTIONARY LEARNING

Abstract. Limitations in imaging systems and the effects of changes in sensing have caused limitation in acquiring high resolution images such as satellite images and magnetic resonance imaging (MRI). Sparsity can reduce the noises and improve the resolution. Super resolution in medical and satellite imagery is essential because low resolution image analysis is very difficult. Sparsity techniques have significant influence on computer vision specially when the main objective is extracting the meaningful information. The success of sparsity is related to the nature of signals such as image and sound which are naturally sparse because they were founded based on Wavelet and Fourier equations. In this research, we proposed a method for restoring a clear image from the related low-resolution parts of both MRI and satellite images. First, we proposed a widespread structure for learning the couple low rank and sparse main characteristic representation. Combined optimization of the nuclear and L1 norms extracts the total low rank formation and the local patterns lodged in the image. In that case the reconstructed image will be more informative and matrix decomposition problem can recover a noisy observation matrix into an approximation of low rank matrix and a second matrix which contains some low dimensional structure. We assumed that by removing the blur and noise from these images, they would be reconstructed in the highest quality. The proposed method was compared with a variety dictionary learning approaches which addressed super resolution problem, such as tensor sparsity, Generative Bayesian and TV based methods. We demonstrated the results of applied method on MRI and satellite images, showing both visual and psnr improvements. Dealing with complex data in best manner shows the robustness of the proposed method.

FOC winding defect detection based on improved texture features and low-rank representation model.

The defect detection of fiber-optic coils (FOCs) plays an important role in the quality control of the FOC production. In order to overcome the problems of poor performance and low reliability of existing methods, this paper provides a solution for winding defect detection of FOCs based on low-rank representation (LRR) technology. First, we design a feature matrix, which represents the image. Then the LRR model is employed to formulate the defect detection task as a problem of low rank and sparse matrix decomposition. Meanwhile, Laplacian regularization is introduced as a smoothness constraint to expand the distance between defect regions and low-rank background. Experiments are performed on a real dataset to verify the algorithm. The results show that the proposed winding defect detection method of FOCs achieves the highest detection accuracy and lowest false alarm rate compared to other methods, verifying the effectiveness of the proposed method.

Low rank and sparse matrix decomposition based multiple targets extraction for forward-looking scanning radar

Forward-looking scanning radar is capable of obtaining the real beam image of terrain in front of the flight platform, and can be used in military and civilian fields, such as sea surface surveillance, search and rescue, etc. However, it is difficult to extract multiple targets from the real beam image due to its poor azimuth resolution. A multiple-target extraction scheme based on low-rank and sparse matrix decomposition is proposed for forward-looking scanning radar. In the scheme, an image with good azimuth resolution is obtained first by the deconvolution preprocessing, and then it is used to map a patch-image. Second, using the low-rank property of the patch-image and the sparse property of the targets of interest, the target extraction is converted into an optimization problem of low-rank and sparse matrix decomposition. Third, the extraction results are obtained by solving this optimization problem using the alternating direction method of multipliers. Finally, simulation and experiment results are given to verify the effectiveness of the proposed scheme.

A generalized health indicator for performance degradation assessment of rolling element bearings based on graph spectrum reconstruction and spectrum characterization

The performance degradation assessment (PDA) of rolling element bearings is a necessary link to ensure the reliability of high-end equipment. However, traditional health indicators (HIs) are not sensitive to early defects, and there are often large local fluctuations in the later stage of degradation. Hence, this paper propose a novel PDA method to obtain HIs with early warning capability and monotone trend. Firstly, an improved graph spectrum reconstruction method is proposed to enhance the characteristics of signals. The random phase space reconstruction strategy is introduced to solve the problem of large-scale graph Laplacian matrix decomposition. Then, the spectrums of the enhanced signals are characterized, namely the high-dimensional degradation features in frequency domain are extracted and smoothed by Kalman filter. Finally, the Laplacian Eigenmaps is used to extract the intrinsic degeneration manifolds from the high-dimensional degradation features as the established HIs. Life cycle degradation data and quantitative failure data are analyzed to verify the effectiveness of the proposed method. Compared with other state-of-art methods, the results show that the HIs established by the proposed method reflect the degradation earlier and have obvious degradation trend. It effectively realizes the mapping between degradation and HI.

Multiscale Low-Rank Spatial Features for Hyperspectral Image Classification

This letter presents a multiscale low-rank decomposition (MSLRD) method to extract multiscale spatial structures from hyperspectral images. The MSLRD assumes that ground objects have divergent characteristics in changing spatial scales. It decomposes each band image into a series of block-wise matrices, where these low-rank blocks take detailed spatial structures at multiple scales. It formulates the low-rank matrix decomposition problem into minimizing the ranks of all block matrices and adopts the alternative direction of the multiplier method to optimize it. Experiments on Indian Pines and Pavia University data sets show that the MSLRD can greatly improve the classification performance of regular classification on spectral features (i.e., all bands) and perform better than five state-of-the-art spatial feature extraction methods.

Fault diagnosis using novel AdaBoost based discriminant locality preserving projection with resamples

Fault diagnosis plays a pivotal role in ensuring the safety of process industries. However, due to the diversity of process faults and the high coupling of fault data, it becomes very difficult to achieve high accuracy in the fault diagnosis of complex industrial processes. To address this concern, in this article, a novel AdaBoost-based discriminant locality preserving projection (DLPP) with resamples (A-DLPPR) model is proposed. The proposed A-DLPPR model has two features: to address the problem of matrix decomposition in DLPP, the bootstrap method is utilized to generate groups of resample data, and to obtain high classification accuracy, the AdaBoost-based classification technique is adopted. Finally, an effective fault diagnosis model using the proposed A-DLPPR model can be established. To validate the effectiveness of the proposed A-DLPPR model, the Tennessee Eastman process (TEP) is selected, and case studies using different kinds of TEP faults are conducted. The simulation results indicate that the proposed A-DLPPR model can achieve higher fault diagnosis accuracy than some other models, which verifies that in the field of complex industrial processes, the proposed A-DLPPR method can be used as an effective model for fault diagnosis.

Error Bounded Foreground and Background Modeling for Moving Object Detection in Satellite Videos

Detecting moving objects from ground-based videos is commonly achieved by using background subtraction techniques. Low-rank matrix decomposition inspires a set of state-of-the-art approaches for this task. It is integrated with structured sparsity regularization to achieve background subtraction in the developed method of Low-rank and Structured Sparse Decomposition (LSD). However, when this method is applied to satellite videos where spatial resolution is poor and targets' contrast to the background is low, its performance is limited as the data no longer fits adequately either the foreground structure or the background model. In this paper, we handle these unexplained data explicitly and address the moving target detection from space as one of the pioneer studies. We propose a technique by extending the decomposition formulation with bounded errors, named Extended Low-rank and Structured Sparse Decomposition (E-LSD). This formulation integrates low-rank background, structured sparse foreground and their residuals in a matrix decomposition problem. We provide an effective solution by introducing an alternative treatment and adopting the direct extension of Alternating Direction Method of Multipliers (ADMM). The proposed E-LSD was validated on two satellite videos, and experimental results demonstrate the improvement in background modeling with boosted moving object detection precision over state-of-the-art methods.

Integrative subspace clustering by common and specific decomposition for applications on cancer subtype identification

BackgroundRecent high throughput technologies have been applied for collecting heterogeneous biomedical omics datasets. Computational analysis of the multi-omics datasets could potentially reveal deep insights for a given disease. Most existing clustering methods by multi-omics data assume strong consistency among different sources of datasets, and thus may lose efficacy when the consistency is relatively weak. Furthermore, they could not identify the conflicting parts for each view, which might be important in applications such as cancer subtype identification.MethodsIn this work, we propose an integrative subspace clustering method (ISC) by common and specific decomposition to identify clustering structures with multi-omics datasets. The main idea of our ISC method is that the original representations for the samples in each view could be reconstructed by the concatenation of a common part and a view-specific part in orthogonal subspaces. The problem can be formulated as a matrix decomposition problem and solved efficiently by our proposed algorithm.ResultsThe experiments on simulation and text datasets show that our method outperforms other state-of-art methods. Our method is further evaluated by identifying cancer types using a colorectal dataset. We finally apply our method to cancer subtype identification for five cancers using TCGA datasets, and the survival analysis shows that the subtypes we found are significantly better than other compared methods.ConclusionWe conclude that our ISC model could not only discover the weak common information across views but also identify the view-specific information.

Low-Rank and Locality Constrained Self-Attention for Sequence Modeling

Self-attention mechanism becomes more and more popular in natural language processing (NLP) applications. Recent studies show the Transformer architecture which relies mainly on the attention mechanism achieves much success on large datasets. But a raised problem is its generalization ability is weaker than CNN and RNN on many moderate-sized datasets. We think the reason can be attributed to its unsuitable inductive bias of the self-attention structure. In this paper, we regard the self-attention as matrix decomposition problem and propose an improved self-attention module by introducing two linguistic constraints: low-rank and locality. We further develop the low-rank attention and band attention to parameterize the self-attention mechanism under the low-rank and locality constraints. Experiments on several real NLP tasks show our model outperforms the vanilla Transformer and other self-attention models on moderate size datasets. Additionally, evaluation on a synthetic task gives us a more detailed understanding of working mechanisms of different architectures.

Novel Pattern Recognition Using Bootstrap-Based Discriminant Locality-Preserving Projection and Its Application to Fault Diagnosis

For the sake of ensuring the safety of complex process industries, accurate fault diagnosis is very important and necessary. Pattern recognition-based techniques have been successfully and widely applied to fault diagnosis. In this article, a novel pattern recognition method using bootstrap-based discriminant locality-preserving projection is proposed. In the proposed bootstrap-based discriminant locality-preserving projection method, a bootstrap is used to resample and construct groups of within-class data. As a result, the matrix decomposition problem of the discriminant locality- preserving projection can be solved with the aid of within-class data. In addition, a novel adjacency graph consisting of a k-nearest-neighbor graph and a k-furthest-neighbor graph is adopted in the proposed model to seek the subspace that best discriminates the different classes, where the distance between class is maximized, while the distance within class is minimized. To verify the performance of the proposed bootstrap-based discriminant locality-preserving projection method, case studies using a two-dimensional synthetic dataset and the Tennessee Eastman process are carried out. Simulation results indicate that compared with some other methods, the proposed Bootstrap-DLPP method can achieve a better visual intuition and a higher accuracy in fault diagnosis.

Hyperspectral Anomaly Detection via Background and Potential Anomaly Dictionaries Construction

In this paper, we propose a new anomaly detection method for hyperspectral images based on two well-designed dictionaries: background dictionary and potential anomaly dictionary. In order to effectively detect an anomaly and eliminate the influence of noise, the original image is decomposed into three components: background, anomalies, and noise. In this way, the anomaly detection task is regarded as a problem of matrix decomposition. Considering the homogeneity of background and the sparsity of anomalies, the low-rank and sparse constraints are imposed in our model. Then, the background and potential anomaly dictionaries are constructed using the background and anomaly priors. For the background dictionary, a joint sparse representation (JSR)-based dictionary selection strategy is proposed, assuming that the frequently used atoms in the overcomplete dictionary tend to be the background. In order to make full use of the prior information of anomalies hidden in the scene, the potential anomaly dictionary is constructed. We define a criterion, i.e., the anomalous level of a pixel, by using the residual calculated in the JSR model within its local region. Then, it is combined with a weighted term to alleviate the influence of noise and background. Experiments show that our proposed anomaly detection method based on potential anomaly and background dictionaries construction can achieve superior results compared with other state-of-the-art methods.

Perfect discrimination of nonorthogonal quantum states with posterior classical partial information

The indistinguishability of non-orthogonal pure states lies at the heart of quantum information processing. Although the indistinguishability reflects the impossibility of measuring complementary physical quantities by a single measurement, we demonstrate that the distinguishability can be perfectly retrieved simply with the help of posterior classical partial information. We demonstrate this by showing an ensemble of non-orthogonal pure states such that a state randomly sampled from the ensemble can be perfectly identified by a single measurement with help of the post-processing of the measurement outcomes and additional partial information about the sampled state, i.e., the label of subensemble from which the state is sampled. When an ensemble consists of two subensembles, we show that the perfect distinguishability of the ensemble with the help of the post-processing can be restated as a matrix-decomposition problem. Furthermore, we give the analytical solution for the problem when both subensembles consist of two states.