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The measurement matrix which plays an important role in compressed sensing has got a lot of attention. However, the existing measurement matrices ignore the energy concentration characteristic of the natural images in the sparse domain, which can help to improve the sensing efficiency and the construction efficiency. Here, the authors propose a simple but efficient measurement matrix based on the Hadamard matrix, named Hadamard-diagonal matrix (HDM). In HDM, the energy conservation in the sparse domain is maximised. In addition, considering the reconstruction performance can be further improved by decreasing the mutual coherence of the measurement matrix, an effective optimisation strategy is adopted in order to reducing the mutual coherence for better reconstruction quality. The authors conduct several experiments to evaluate the performance of HDM and the effectiveness of optimisation algorithm. The experimental results show that HDM performs better than other popular measurement matrices, and the optimisation algorithm can improve the performance of not only the HDM but also the other popular measurement matrices.

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad scientific community. The treatment of eigenvectors is more complicated, with a perturbation theory that is not so well known outside a community of specialists. We give two different proofs of the main eigenvector perturbation theorem. The first, a block-diagonalization technique inspired by the numerical linear algebra research community and based on the implicit function theorem, has apparently not appeared in the literature in this form. The second, based on complex function theory and on eigenprojectors, as is standard in analytic perturbation theory, is a simplified version of well-known results in the literature. The second derivation uses a convenient normalization of the right and left eigenvectors defined in terms of the associated eigenprojector, but although this dates back to the 1950s, it is rarely discussed in the literature. We then show how the eigenvector perturbation theory is easily extended to handle other normalizations that are often used in practice. We also explain how to verify the perturbation results computationally. We conclude with some remarks about difficulties introduced by multiple eigenvalues and give references to work on perturbation of invariant subspaces corresponding to multiple or clustered eigenvalues. Throughout the paper we give extensive bibliographic commentary and references for further reading.

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or N-way array. Decompositions of higher-order tensors (i.e., N-way arrays with N ≥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

We present a path algorithm for the generalized lasso problem. This problem penalizes the ℓ 1 norm of a matrix D times the coefficient vector, and has a wide range of applications, dictated by the choice of D. Our algorithm is based on solving the dual of the generalized lasso, which greatly facilitates computation of the path. For D = I (the usual lasso), we draw a connection between our approach and the well-known LARS algorithm. For an arbitrary D, we derive an unbiased estimate of the degrees of freedom of the generalized lasso fit. This estimate turns out to be quite intuitive in many applications.

In the paper I give a brief review of the basic idea and some history and then discuss some developments since the original paper on regression shrinkage and selection via the lasso. [PUBLICATION ABSTRACT]

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k)) floating-point operations (flops) in contrast to O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multi-processor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data.