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Hyperspectral imaging captures detailed spectral data for remote sensing. However, due to the limited spatial resolution of hyperspectral sensors, each pixel of a hyperspectral image (HSI) may contain information from multiple materials. Although the hyperspectral unmixing (HU) process involves estimating endmembers, identifying pure spectral components, and estimating pixel abundances, existing algorithms mostly focus on just one or two tasks. Blind source separation (BSS) based on nonnegative matrix factorization (NMF) algorithms identify endmembers and their abundances at each pixel of HSI simultaneously. Although they perform well, the factorization results are unstable, require high computational costs, and are difficult to interpret from the original HSI. CUR matrix decomposition selects specific columns and rows from a dataset to represent it as a product of three small submatrices, resulting in interpretable low-rank factorization. In this paper, we propose a new blind HU framework based on CUR factorization called CUR-HU that performs the entire HU process by exploiting the low-rank structure of given HSIs. CUR-HU incorporates several techniques to perform the HU process with a performance comparable to state-of-the-art methods but with higher computational efficiency. We adopt a deterministic sampling method to select the most informative pixels and spectrum components in HSIs. We use an incremental QR decomposition method to reduce computation complexity and estimate the number of endmembers. Various experiments on synthetic and real HSIs are conducted to evaluate the performance of CUR-HU. CUR-HU performs comparably to state-of-the-art methods for estimating the number of endmembers and abundance maps, but it outperforms other methods for estimating the endmembers and the computational efficiency. It has a 9.4 to 249.5 times speedup over different methods for different real HSIs.

Multi-focus image fusion integrates images from multiple focus regions of the same scene in focus to produce a fully focused image. However, the accurate retention of the focused pixels to the fusion result remains a major challenge. This study proposes a multi-focus image fusion algorithm based on Hessian matrix decomposition and salient difference focus detection, which can effectively retain the sharp pixels in the focus region of a source image. First, the source image was decomposed using a Hessian matrix to obtain the feature map containing the structural information. A focus difference analysis scheme based on the improved sum of a modified Laplacian was designed to effectively determine the focusing information at the corresponding positions of the structural feature map and source image. In the process of the decision-map optimization, considering the variability of image size, an adaptive multiscale consistency verification algorithm was designed, which helped the final fused image to effectively retain the focusing information of the source image. Experimental results showed that our method performed better than some state-of-the-art methods in both subjective and quantitative evaluation.

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification and is intractable to solve in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components by minimizing a linear combination of the [cursive l]1 norm and the nuclear norm of the components. We develop a notion of rank-sparsity incoherence, expressed as an uncertainty principle between the sparsity pattern of a matrix and its row and column spaces, and we use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature with the tangent spaces to the algebraic varieties of sparse and low-rank matrices playing a prominent role. When the sparse and low-rank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems. [PUBLICATION ABSTRACT]

As the existing processing algorithms for LiDAR echo decomposition are time-consuming, this paper proposes an FPGA-based improved Gaussian full-waveform decomposition method. The proposed FPGA architecture consists of three modules: (i) a pre-processing module, which is used to pipeline data reading and Gaussian filtering, (ii) the inflection point coordinate solution module, applied to the second-order differential operation and to calculate inflection point coordinates, and (iii) the Gaussian component parameter solution and echo component positioning module, which is utilized to calculate the Gaussian component and echo time parameters. Finally, two LiDAR datasets, covering the Congo and Antarctic regions, are used to verify the accuracy and speed of the proposed method. The experimental results show that (i) the accuracy of the FPGA-based processing is equivalent to that of PC-based processing, and (ii) the processing speed of the FPGA-based processing is 292 times faster than that of PC-based processing.

Recently, a growing number of biological research and scientific experiments have demonstrated that microRNA (miRNA) affects the development of human complex diseases. Discovering miRNA-disease associations plays an increasingly vital role in devising diagnostic and therapeutic tools for diseases. However, since uncovering associations via experimental methods is expensive and time-consuming, novel and effective computational methods for association prediction are in demand. In this study, we developed a computational model of Matrix Decomposition and Heterogeneous Graph Inference for miRNA-disease association prediction (MDHGI) to discover new miRNA-disease associations by integrating the predicted association probability obtained from matrix decomposition through sparse learning method, the miRNA functional similarity, the disease semantic similarity, and the Gaussian interaction profile kernel similarity for diseases and miRNAs into a heterogeneous network. Compared with previous computational models based on heterogeneous networks, our model took full advantage of matrix decomposition before the construction of heterogeneous network, thereby improving the prediction accuracy. MDHGI obtained AUCs of 0.8945 and 0.8240 in the global and the local leave-one-out cross validation, respectively. Moreover, the AUC of 0.8794+/-0.0021 in 5-fold cross validation confirmed its stability of predictive performance. In addition, to further evaluate the model's accuracy, we applied MDHGI to four important human cancers in three different kinds of case studies. In the first type, 98% (Esophageal Neoplasms) and 98% (Lymphoma) of top 50 predicted miRNAs have been confirmed by at least one of the two databases (dbDEMC and miR2Disease) or at least one experimental literature in PubMed. In the second type of case study, what made a difference was that we removed all known associations between the miRNAs and Lung Neoplasms before implementing MDHGI on Lung Neoplasms. As a result, 100% (Lung Neoplasms) of top 50 related miRNAs have been indexed by at least one of the three databases (dbDEMC, miR2Disease and HMDD V2.0) or at least one experimental literature in PubMed. Furthermore, we also tested our prediction method on the HMDD V1.0 database to prove the applicability of MDHGI to different datasets. The results showed that 50 out of top 50 miRNAs related with the breast neoplasms were validated by at least one of the three databases (HMDD V2.0, dbDEMC, and miR2Disease) or at least one experimental literature.

Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of \"components.\" Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and/or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an$m\\times n$matrix$A$and a rank parameter$k$ . In our first algorithm,$C$is chosen, and we let$A'=CC^+A$ , where$C^+$is the Moore-Penrose generalized inverse of$C$ . In our second algorithm$C$ ,$U$ ,$R$are chosen, and we let$A'=CUR$ . ( $C$and$R$are matrices that consist of actual columns and rows, respectively, of$A$ , and$U$is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least$1-\\delta$ ,$\\|A-A'\\|_F\\leq(1+\\epsilon)\\,\\|A-A_k\\|_F$ , where$A_k$is the \"best\" rank- $k$approximation provided by truncating the SVD of$A$ , and where$\\|X\\|_F$is the Frobenius norm of the matrix$X$ . The number of columns of$C$and rows of$R$is a low-degree polynomial in$k$ ,$1/\\epsilon$ , and$\\log(1/\\delta)$ . Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants of these matrix decompositions over the last ten years. However, our two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, it was not even known whether such matrix decompositions exist. Both of our algorithms are simple and they take time of the order needed to approximately compute the top$k$singular vectors of$A$ . The technical crux of our analysis is a novel, intuitive sampling method we introduce in this paper called \"subspace sampling.\" In subspace sampling, the sampling probabilities depend on the Euclidean norms of the rows of the top singular vectors. This allows us to obtain provable relative-error guarantees by deconvoluting \"subspace\" information and \"size-of- $A$ \" information in the input matrix. This technique is likely to be useful for other matrix approximation and data analysis problems.

This paper studies the possibilities of the linear matrix inequality characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real-case analog, such studies were conducted in Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246–267]. In this paper it is shown that stronger results can be obtained for the complex Hermitian case. In particular, we show that the matrix rank-one decomposition result of Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246–267] can be strengthened for the complex Hermitian matrices. As a consequence, it is possible to characterize several new matrix co-positive cones (over specific domains) by means of linear matrix inequality. As examples of the potential application of the new rank-one decomposition result, we present an upper bound on the lowest rank among all the optimal solutions for a standard complex semidefinite programming (SDP) problem, and offer alternative proofs for a result of Hausdorff [Hausdorff, F. 1919. Der Wertvorrat einer Bilinearform. Mathematische Zeitschrift 3 314–316] and a result of Brickman [Brickman, L. 1961. On the field of values of a matrix. Proc. Amer. Math. Soc. 12 61–66] on the joint numerical range.

We introduce the Hadamard decomposition problem in the context of data analysis. The problem is to represent exactly or approximately a given matrix as the Hadamard (or element-wise) product of two or more low-rank matrices. The motivation for this problem comes from situations where the input matrix has a multiplicative structure. The Hadamard decomposition has potential for giving more succint but equally accurate representations of matrices when compared with the gold-standard of singular value decomposition (svd). Namely, the Hadamard product of two rank-h matrices can have rank as high as h2. We study the computational properties of the Hadamard decomposition problem and give gradient-based algorithms for solving it approximately. We also introduce a mixed model that combines svd and Hadamard decomposition. We present extensive empirical results comparing the approximation accuracy of the Hadamard decomposition with that of the svd using the same number of basis vectors. The results demonstrate that the Hadamard decomposition is competitive with the svd and, for some datasets, it yields a clearly higher approximation accuracy, indicating the presence of multiplicative structure in the data.

We study the problem of decomposing the Hessian matrix of a mixed integer convex quadratic program (MICQP) into the sum of positive semidefinite 2 * 2 matrices. Solving this problem enables the use of perspective reformulation techniques for obtaining strong lower bounds for MICQPs with semicontinuous variables but a nonseparable objective function. An explicit formula is derived for constructing 2 * 2 decompositions when the underlying matrix is weakly scaled diagonally dominant, and necessary and sufficient conditions are given for the decomposition to be unique. For matrices lying outside this class, two exact semidefinite programming approaches and an efficient heuristic are developed for finding approximate decompositions. We present preliminary results on the bound strength of a 2 * 2 perspective reformulation for the portfolio optimization problem, showing that, for some classes of instances, the use of 2 * 2 matrices can significantly improve the quality of the bound with respect to the best previously known approach, although at a possibly high computational cost.