Matrix Decomposition Algorithm Research Articles

Exploiting Continuity/Discontinuity of Basis Vectors in Spectrogram Decomposition for Harmonic-Percussive Sound Separation

In this paper, we present a novel method for harmonic-percussive sound separation (HPSS) by exploiting continuity/discontinuity properties in a matrix decomposition framework. It is widely accepted in the HPSS research that the harmonic and percussive components have anisotropic characteristics: The spectrum of the harmonic components and the time activation of the percussive components are sparse, whereas the spectrum of the percussive components and the time activation of the harmonic components are smooth. However, conventional methods fail to fully utilize the characteristics leading to suboptimal performance. Based on the observations that not the degree of sparseness but the degree of fluctuation is an accurate measure for distinguishing the harmonic and percussive components, we propose a novel HPSS algorithm by incorporating the continuity control in the iterative update formula of the matrix decomposition algorithm. In doing so, we first utilize probabilistic latent component analysis with Dirichlet prior, and later reformulate the algorithm in the nonnegative matrix factorization framework to reduce the computational cost. The comparative evaluation results show that the proposed method outperforms conventional methods in terms of both objective and subjective evaluation.

A Parallel Multiclassification Algorithm for Big Data Using an Extreme Learning Machine.

As data sets become larger and more complicated, an extreme learning machine (ELM) that runs in a traditional serial environment cannot realize its ability to be fast and effective. Although a parallel ELM (PELM) based on MapReduce to process large-scale data shows more efficient learning speed than identical ELM algorithms in a serial environment, some operations, such as intermediate results stored on disks and multiple copies for each task, are indispensable, and these operations create a large amount of extra overhead and degrade the learning speed and efficiency of the PELMs. In this paper, an efficient ELM based on the Spark framework (SELM), which includes three parallel subalgorithms, is proposed for big data classification. By partitioning the corresponding data sets reasonably, the hidden layer output matrix calculation algorithm, matrix decomposition algorithm, and matrix decomposition algorithm perform most of the computations locally. At the same time, they retain the intermediate results in distributed memory and cache the diagonal matrix as broadcast variables instead of several copies for each task to reduce a large amount of the costs, and these actions strengthen the learning ability of the SELM. Finally, we implement our SELM algorithm to classify large data sets. Extensive experiments have been conducted to validate the effectiveness of the proposed algorithms. As shown, our SELM achieves an speedup on a cluster with ten nodes, and reaches a speedup with 15 nodes, an speedup with 20 nodes, a speedup with 25 nodes, a speedup with 30 nodes, and a speedup with 35 nodes.

GoDec+: Fast and Robust Low-Rank Matrix Decomposition Based on Maximum Correntropy.

GoDec is an efficient low-rank matrix decomposition algorithm. However, optimal performance depends on sparse errors and Gaussian noise. This paper aims to address the problem that a matrix is composed of a low-rank component and unknown corruptions. We introduce a robust local similarity measure called correntropy to describe the corruptions and, in doing so, obtain a more robust and faster low-rank decomposition algorithm: GoDec+. Based on half-quadratic optimization and greedy bilateral paradigm, we deliver a solution to the maximum correntropy criterion (MCC)-based low-rank decomposition problem. Experimental results show that GoDec+ is efficient and robust to different corruptions including Gaussian noise, Laplacian noise, salt & pepper noise, and occlusion on both synthetic and real vision data. We further apply GoDec+ to more general applications including classification and subspace clustering. For classification, we construct an ensemble subspace from the low-rank GoDec+ matrix and introduce an MCC-based classifier. For subspace clustering, we utilize GoDec+ values low-rank matrix for MCC-based self-expression and combine it with spectral clustering. Face recognition, motion segmentation, and face clustering experiments show that the proposed methods are effective and robust. In particular, we achieve the state-of-the-art performance on the Hopkins 155 data set and the first 10 subjects of extended Yale B for subspace clustering.

High Dimensional Low Rank Plus Sparse Matrix Decomposition

This paper is concerned with the problem of low rank plus sparse matrix decomposition for big data. Conventional algorithms for matrix decomposition use the entire data to extract the low-rank and sparse components, and are based on optimization problems with complexity that scales with the dimension of the data, which limits their scalability. Furthermore, existing randomized approaches mostly rely on uniform random sampling, which is quite inefficient for many real world data matrices that exhibit additional structures (e.g. clustering). In this paper, a scalable subspace-pursuit approach that transforms the decomposition problem to a subspace learning problem is proposed. The decomposition is carried out using a small data sketch formed from sampled columns/rows. Even when the data is sampled uniformly at random, it is shown that the sufficient number of sampled columns/rows is roughly O(r\mu), where \mu is the coherency parameter and r the rank of the low rank component. In addition, adaptive sampling algorithms are proposed to address the problem of column/row sampling from structured data. We provide an analysis of the proposed method with adaptive sampling and show that adaptive sampling makes the required number of sampled columns/rows invariant to the distribution of the data. The proposed approach is amenable to online implementation and an online scheme is proposed.

A tensor product finite element method for the diffraction grating problem with transparent boundary conditions

We consider the diffraction grating problem in optics, which has been modeled by a boundary value problem governed by a Helmholtz equation with transparent boundary conditions. A tensor-product finite element method is proposed to numerically solve the problem. An FFT-based matrix decomposition algorithm is developed to solve the linear system arising in the vertically layered medium case, which can be used as a preconditioning technique for the general case. Numerical examples are presented to illustrate the accuracy and efficiency of the method.

Conformal Mapping for the Efficient Solution of Poisson Problems with the Kansa-RBF Method

We consider the solution of Poisson Dirichlet problems in simply-connected irregular domains. These domains are conformally mapped onto the unit disk and the resulting Poisson Dirichlet problems are solved efficiently using a Kansa-radial basis function (RBF) method with a matrix decomposition algorithm (MDA). In a similar way, we treat Poisson Dirichlet and Poisson Dirichlet---Neumann problems in doubly-connected domains. These domains are mapped onto annular domains by a conformal mapping and the resulting Poisson Dirichlet and Poisson Dirichlet---Neumann problems are solved efficiently using a Kansa-RBF MDA. Several examples demonstrating the applicability of the proposed technique are presented.

The plane waves method for axisymmetric Helmholtz problems

The plane waves method is employed for the solution of Dirichlet and Neumann boundary value problems for the homogeneous Helmholtz equation in two- and three-dimensional domains possessing radial symmetry. The appropriate selection of collocation points and unitary direction vectors in the method leads to circulant and block circulant coefficient matrices in two and three dimensions, respectively. We propose efficient matrix decomposition algorithms which make use of fast Fourier transforms for the solution of the systems resulting from such a discretization. In conjunction with the method of particular solutions, the method is extended to the solution of inhomogeneous axisymmetric Helmholtz problems. Several numerical examples are presented.

Data Flow Transformation for Energy-Efficient Implementation of Givens Rotation--Based QRD

QR decomposition (QRD), a matrix decomposition algorithm widely used in embedded application domain, can be realized in a large number of valid processing sequences that differ significantly in the number of memory accesses and computations, and hence the overall implementation energy. With modern low-power embedded processors evolving toward register files with wide memory interfaces and vector functional units (FUs), data flow in these algorithms needs to be carefully devised to efficiently utilize the costly wide memory accesses and the vector FUs. In this article, we present an energy-efficient data flow transformation strategy for the Givens rotation--based QRD.

Research Spotlights

This issue's Research Spotlights section contains three papers. First, “Pairwise Compatibility Graphs: A Survey," by Tiziana Calamoneri and Blerina Sinaimeri, describes computational graph theory methods as applied to computational biology. The evolutionary ancestral relationships of species can be described using a tree-like graph structure known as a phylogenetic tree. Using optimization techniques, the goal is to reconstruct the best possible tree structure starting from a set of biological data. However, the data sets are large and the phylogenetic tree reconstruction problem is known to be NP-hard. This article presents an approach using carefully chosen subsets of the original data set. The work is of particular interest to a reader interested in graph theory or computational biology. The second article in this section, “Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions," by Yuji Nakatsukasa and Roland W. Freund, describes a new algorithm for matrix decomposition. The paper develops a high order method using functions of Zolotarev (1877) to calculate two standard matrix decompositions: singular value and symmetric eigenvalue decompositions. The method is designed to be well suited for parallel computing. This article gives a self-contained description of the method and its convergence. It will appeal to those interested in either the theory of numerical linear algebra or its application for large-scale matrix decompositions. The third paper, “Optimal Mixing Enhancement by Local Perturbation," by Gary Froyland, Cecilia González-Tokman, and Thomas M. Watson, develops optimal methods for applying local perturbations in order to most effectively increase the process of mixing in a dynamical system. The approach is a systematic way of turning a dynamical system into a convex optimization problem. This paper will appeal to a broad set of readers, as the control of fluid mixing not only is of theoretical interest but also has wide-ranging applications on many different length scales within engineering and natural sciences. For example, it is important in the speedup of industrial chemical mixing, the prevention of pollutant spread in oceanographic and atmospheric flow, and the control of microfluidics.

Kansa-RBF Algorithms for Elliptic Problems in Axisymmetric Domains

We employ a Kansa-radial basis function method for the numerical solution of elliptic boundary value problems in three-dimensional axisymmetric domains. We consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy--Navier equations. Appropriate discretizations lead to linear systems, the coefficient matrices of which possess block circulant structures. These systems are then solved efficiently by means of matrix decomposition algorithms and fast Fourier transforms. Methods for choosing appropriate values of the shape parameter are also proposed. The effectiveness of the proposed algorithms is demonstrated by considering several numerical examples.

Graph regularized and sparse nonnegative matrix factorization with hard constraints for data representation

Nonnegative Matrix Factorization (NMF) as a popular technique for finding parts-based, linear representations of nonnegative data has been successfully applied in a wide range of applications. This is because it can provide components with physical meaning and interpretations, which is consistent with the psychological intuition of combining parts to form whole. For practical classification tasks, NMF ignores both the local geometry of data and the discriminative information of different classes. In addition, existing research results demonstrate that leveraging sparseness can greatly enhance the ability of the learning parts. Motivated by these advances aforementioned, we propose a novel matrix decomposition algorithm, called Graph regularized and Sparse Non-negative Matrix Factorization with hard Constraints (GSNMFC). It attempts to find a compact representation of the data so that further learning tasks can be facilitated. The proposed GSNMFC jointly incorporates a graph regularizer and hard prior label information as well as sparseness constraint as additional conditions to uncover the intrinsic geometrical and discriminative structures of the data space. The corresponding update solutions and the convergence proofs for the optimization problem are also given in detail. Experimental results demonstrate the effectiveness of our algorithm in comparison to the state-of-the-art approaches through a set of evaluations.

Data-Sparse Matrix Decomposition Algorithm for Mixed Finite-Element Time-Domain Solution of Maxwell Equations

In this article, an efficient algorithm is presented for the mixed finite-element time-domain solution of Maxwell equations. The implicit finite-element time-domain method based on the Crank–Nicolson time integration is unconditionally stable but involves solving a linear system at each time step. A data-sparse matrix decomposition algorithm combined with the nested dissection technique is developed to solve the large linear system. Based on the hierarchical matrix (H-matrix) formatted arithmetic, this algorithm can reduce the computational costs to be almost linear. Moreover, an iterative improvement method is introduced to further improve the solution performance. Numerical results demonstrate the effectiveness of the proposed method for the analysis of microwave circuits.

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

In this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

Covariance differencing‐based matrix decomposition for coherent sources localisation in bi‐static multiple‐input–multiple‐output radar

In this study, a covariance differencing-based matrix decomposition algorithm is proposed for locating coherent sources under spatially coloured noise in bi-static multiple-input–multiple-output (MIMO) radar. The method contains three steps. First, the covariance differencing technique is employed to eliminate sensor noise, especially the spatially coloured noise. Second, a block Toeplitz or block Hankel matrix is constructed for decorrelation with the covariance differenced matrix. The forward-only, backward-only and combined forward-backward block Toeplitz/Hankel matrix constructions are defined, respectively. Third, unitary estimation of signal parameters by rotational invariance techniques (ESPRIT) algorithm is applied to estimate directions-of-departure (DODs) and directions-of-arrival (DOAs) of sources. The proposed algorithm offers several advantages. First, it is more robust and provides better estimation performance than other methods. Then, the coloured noise problem is overcome in a simple and effective way. Further, the computational load is comparatively low. Simulation results demonstrate the validity of the proposed algorithm.

Switch cost and packet delay tradeoff in data center networks with switch reconfiguration overhead

Cost minimization is a major concern in data center networks (DCNs). Existing DCNs generally adopt Clos network with crossbar middle switches to achieve non-blocking data switching among the servers, and the number of middle switches is proportional to the number of ports of the aggregation switches in a fixed manner. Besides, reconfiguration overhead of the switches is generally ignored, which may contradict the engineering practice. In this paper, we consider batch scheduling based packet switching in DCNs with reconfiguration overhead at each middle switch, which inevitably leads to packet delay. With existing state-of-the-art traffic matrix decomposition algorithms, we can generate a set of permutations, each of which stands for the configuration of a middle switch. By reconfiguring each middle switch to fulfill multiple configurations in parallel with others, we reveal that a tradeoff exists between packet delay and switch cost (denoted by the number of middle switches), while performance guaranteed switching with bounded packet delay can be achieved without any packet loss. Based on the tradeoff, we can minimize the number of middle switches (under a given packet delay bound) and an overall cost metric (by translating delay into a comparable cost factor), as well as formulating criteria for choosing a proper matrix decomposition algorithm. This provides a flexible way to reduce the number of middle switches by slightly enlarging the packet delay bound.

A fast direct solver for a fourth order finite difference scheme for Poisson’s equation on the unit disc in polar coordinates

We present a fourth order finite difference scheme for solving Poisson's equation on the unit disc in polar coordinates. We use a half-point shift in the r direction to avoid approximating the solution at r = 0. We derive our scheme from analysis of the local truncation error of the standard second order finite difference scheme. The resulting linear system is solved very efficiently (with cost almost proportional to the number of unknowns) using a matrix decomposition algorithm with fast Fourier transforms.

A Kansa-Radial Basis Function Method for Elliptic Boundary Value Problems in Annular Domains

We employ a Kansa-radial basis function (RBF) method for the numerical solution of elliptic boundary value problems in annular domains. This discretization leads, with an appropriate selection of collocation points and for any choice of RBF, to linear systems in which the matrices possess block circulant structures. These linear systems can be solved efficiently using matrix decomposition algorithms and fast Fourier transforms. A suitable value for the shape parameter in the various RBFs used is found using the leave-one-out cross validation algorithm. In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy---Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples.

A Tensor-Based Approach for Big Data Representation and Dimensionality Reduction

Variety and veracity are two distinct characteristics of large-scale and heterogeneous data. It has been a great challenge to efficiently represent and process big data with a unified scheme. In this paper, a unified tensor model is proposed to represent the unstructured, semistructured, and structured data. With tensor extension operator, various types of data are represented as subtensors and then are merged to a unified tensor. In order to extract the core tensor which is small but contains valuable information, an incremental high order singular value decomposition (IHOSVD) method is presented. By recursively applying the incremental matrix decomposition algorithm, IHOSVD is able to update the orthogonal bases and compute the new core tensor. Analyzes in terms of time complexity, memory usage, and approximation accuracy of the proposed method are provided in this paper. A case study illustrates that approximate data reconstructed from the core set containing 18% elements can guarantee 93% accuracy in general. Theoretical analyzes and experimental results demonstrate that the proposed unified tensor model and IHOSVD method are efficient for big data representation and dimensionality reduction.

3D Restructuring and Splicing Technology of Shape Space Frame

The 3D Shape Measuring Technology for physical objects has wide application in fields covering automatic detection, physical object profile modelling, automatic production, computer aided design/manufacturing and quality control, etc. Through the matching relations of several feature points between the partial 3D scanning point cloud data and the space frame, the transformation relations between different coordinate systems can be inferred. As for the inferring of the transformation parameters R and T, the SVD matrix decomposition algorithm has been used to acquire a better precision solution. According to the transformation relation, after registering the partial 3D scanning point cloud data and the space frame, through splicing the partial scanning point cloud data and the space frame data, the detection across the whole field on large-area objects can be realized, which will effectively increase the precision of three-dimensional measuring of large-area objects.

Matrix decomposition algorithms for arbitrary order [formula omitted] tensor product finite element systems

Matrix decomposition algorithms (MDAs) are fast direct methods for the solution of systems of linear algebraic equations which arise in the approximation of Poisson’s equation on the unit square using various techniques such as finite difference, spline collocation and spectral methods. The attraction of MDAs is that they employ fast Fourier transforms and require O(N2logN) operations on an N×N uniform partition of the unit square. In this paper, MDAs are formulated for the solution of the finite element Galerkin equations arising when spaces of C0 piecewise polynomials of degree k≥3 are employed. Results of numerical experiments exhibit the expected optimal global convergence rates and superconvergence phenomena.