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In-crowd Algorithm
The in-crowd algorithm is a numerical method for solving basis pursuit denoising quickly; faster than any other algorithm for large, sparse problems.See ''The In-Crowd Algorithm for Fast Basis Pursuit Denoising'', IEEE Trans Sig Proc 59 (10), Oct 1 2011, pp. 4595 - 4605 demo MATLAB code availabl/ref> This algorithm is an active set method, which minimizes iteratively sub-problems of the global basis pursuit denoising: \min_x \frac\, y-Ax\, ^2_2+\lambda\, x\, _1. where y is the observed signal, x is the sparse signal to be recovered, Ax is the expected signal under x, and \lambda is the regularization parameter trading off signal fidelity and simplicity. The simplicity is here measured using the sparsity of the solution x, measure through its \ell_1-norm. The active set strategies are very efficient in this context as only few coefficient are expected to be non-zero. Thus, if they can be identified, solving the problem restricted to these coefficients yield the solution. Here, the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Pursuit Denoising
In applied mathematics and statistics, basis pursuit denoising (BPDN) refers to a mathematical optimization problem of the form : \min_x \left(\frac \, y - Ax\, ^2_2 + \lambda \, x\, _1\right), where \lambda is a parameter that controls the trade-off between sparsity and reconstruction fidelity, x is an N \times 1 solution vector, y is an M \times 1 vector of observations, A is an M \times N transform matrix and M < N. This is an instance of . Some authors refer to basis pursuit denoising as the following closely related problem: : which, for any given , is equivalent to the unconstrained formulation for some (usually unknown ''a priori'') value of . The two problems are quite similar. In practice, the unconstrained fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. , MATLAB has more than four million users worldwide. They come from various backgrounds of engineering, science, and economics. , more than 5000 global colleges and universities use MATLAB to support instruction and research. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was base ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Active Set Method
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Duality Gap
In optimization problems in applied mathematics, the duality gap is the difference between the primal and dual solutions. If d^* is the optimal dual value and p^* is the optimal primal value then the duality gap is equal to p^* - d^*. This value is always greater than or equal to 0 (for minimization problems). The duality gap is zero if and only if strong duality holds. Otherwise the gap is strictly positive and weak duality holds. In general given two dual pairs separated locally convex spaces \left(X,X^*\right) and \left(Y,Y^*\right). Then given the function f: X \to \mathbb \cup \, we can define the primal problem by :\inf_ f(x). \, If there are constraint conditions, these can be built into the function f by letting f = f + I_\text where I is the indicator function. Then let F: X \times Y \to \mathbb \cup \ be a perturbation function such that F(x,0) = f(x). The ''duality gap'' is the difference given by :\inf_ (x,0)- \sup_ F^*(0,y^*)/math> where F^* is the convex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
