scholarly journals Linear Program Relaxation of Sparse Nonnegative Recovery in Compressive Sensing Microarrays

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Linxia Qin ◽  
Naihua Xiu ◽  
Lingchen Kong ◽  
Yu Li
Keyword(s):  
Compressive Sensing ◽  
Unique Solution ◽  
Linear Equations ◽  
Group Testing ◽  
Extreme Points ◽  
System Of Linear Equations ◽  
Underdetermined System ◽  
The Common ◽  
Property Condition ◽  
Sparsest Solutions

Compressive sensing microarrays (CSM) are DNA-based sensors that operate using group testing and compressive sensing principles. Mathematically, one can cast the CSM as sparse nonnegative recovery (SNR) which is to find the sparsest solutions subjected to an underdetermined system of linear equations and nonnegative restriction. In this paper, we discuss thel1relaxation of the SNR. By defining nonnegative restricted isometry/orthogonality constants, we give a nonnegative restricted property condition which guarantees that the SNR and thel1relaxation share the common unique solution. Besides, we show that any solution to the SNR must be one of the extreme points of the underlying feasible set.

scholarly journals Efficient LED-SAC Sparse Estimator Using Fast Sequential Adaptive Coordinate-Wise Optimization (LED-2SAC)

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
T. Yousefi Rezaii ◽  
S. Beheshti ◽  
M. A. Tinati
Keyword(s):  
Objective Function ◽  
Linear Equations ◽  
Optimization Criterion ◽  
Reconstruction Method ◽  
Solution Path ◽  
System Of Linear Equations ◽  
Processing Application ◽  
Underdetermined System ◽  
Sequential Solution ◽  
Better Than

Solving the underdetermined system of linear equations is of great interest in signal processing application, particularly when the underlying signal to be estimated is sparse. Recently, a new sparsity encouraging penalty function is introduced as Linearized Exponentially Decaying penalty, LED, which results in the sparsest solution for an underdetermined system of equations subject to the minimization of the least squares loss function. A sequential solution is available for LED-based objective function, which is denoted by LED-SAC algorithm. This solution, which aims to sequentially solve the LED-based objective function, ignores the sparsity of the solution. In this paper, we present a new sparse solution. The new method benefits from the sparsity of the signal both in the optimization criterion (LED) and its solution path, denoted by Sparse SAC (2SAC). The new reconstruction method denoted by LED-2SAC (LED-Sparse SAC) is consequently more efficient and considerably fast compared to the LED-SAC algorithm, in terms of adaptability and convergence rate. In addition, the computational complexity of both LED-SAC and LED-2SAC is shown to be of order𝒪d2, which is better than the other batch solutions like LARS. LARS algorithm has complexity of order𝒪d3+nd2, wheredis the dimension of the sparse signal andnis the number of observations.

scholarly journals On the number of Hadamard matrices via anti-concentration

2021 ◽  
pp. 1-23
Author(s):  
Asaf Ferber ◽  
Vishesh Jain ◽  
Yufei Zhao
Keyword(s):  
Matrix Theory ◽  
Linear Equations ◽  
Hadamard Matrices ◽  
Small Subset ◽  
Concentration Inequality ◽  
System Of Linear Equations ◽  
Number Of Solutions ◽  
Underdetermined System ◽  
Concentration Problem ◽  
Novel Approach

Abstract Many problems in combinatorial linear algebra require upper bounds on the number of solutions to an underdetermined system of linear equations $Ax = b$ , where the coordinates of the vector x are restricted to take values in some small subset (e.g. $\{\pm 1\}$ ) of the underlying field. The classical ways of bounding this quantity are to use either a rank bound observation due to Odlyzko or a vector anti-concentration inequality due to Halász. The former gives a stronger conclusion except when the number of equations is significantly smaller than the number of variables; even in such situations, the hypotheses of Halász’s inequality are quite hard to verify in practice. In this paper, using a novel approach to the anti-concentration problem for vector sums, we obtain new Halász-type inequalities that beat the Odlyzko bound even in settings where the number of equations is comparable to the number of variables. In addition to being stronger, our inequalities have hypotheses that are considerably easier to verify. We present two applications of our inequalities to combinatorial (random) matrix theory: (i) we obtain the first non-trivial upper bound on the number of $n\times n$ Hadamard matrices and (ii) we improve a recent bound of Deneanu and Vu on the probability of normality of a random $\{\pm 1\}$ matrix.

scholarly journals Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

2021 ◽  
Author(s):  
David Ek ◽  
Anders Forsgren
Keyword(s):  
Approximate Solution ◽  
Interior Point ◽  
Interior Point Methods ◽  
Linear Equations ◽  
Approximate Solutions ◽  
System Of Nonlinear Equations ◽  
Intermediate Step ◽  
System Of Linear Equations ◽  
Constrained Nonlinear Optimization ◽  
Asymptotic Error

AbstractThe focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

scholarly journals Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
A. Khalid ◽  
M. N. Naeem ◽  
P. Agarwal ◽  
A. Ghaffar ◽  
Z. Ullah ◽  
...  
Keyword(s):  
Numerical Approximation ◽  
Analytic Solution ◽  
Linear Equations ◽  
Computational Cost ◽  
Spline Method ◽  
System Of Linear Equations ◽  
Approximation Solution ◽  
B Spline ◽  
Novel Technique ◽  
Derivatives Of

AbstractIn the current paper, authors proposed a computational model based on the cubic B-spline method to solve linear 6th order BVPs arising in astrophysics. The prescribed method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 6th order BVPs using cubic B-spline, but it also describes the estimated derivatives of 1st order to 6th order of the analytic solution at the same time. This novel technique has lesser computational cost than numerous other techniques and is second order convergent. To show the efficiency of the proposed method, four numerical examples have been tested. The results are described using error tables and graphs and are compared with the results existing in the literature.

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