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      Analyzing Weighted <formula formulatype="inline"> <tex Notation="TeX">$\ell_1$</tex></formula> Minimization for Sparse Recovery With Nonuniform Sparse Models

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          Enhancing Sparsity by Reweighted ℓ 1 Minimization

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            Is Open Access

            Decoding by Linear Programming

            This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector \(f \in \R^n\) from corrupted measurements \(y = A f + e\). Here, \(A\) is an \(m\) by \(n\) (coding) matrix and \(e\) is an arbitrary and unknown vector of errors. Is it possible to recover \(f\) exactly from the data \(y\)? We prove that under suitable conditions on the coding matrix \(A\), the input \(f\) is the unique solution to the \(\ell_1\)-minimization problem (\(\|x\|_{\ell_1} := \sum_i |x_i|\)) \[ \min_{g \in \R^n} \| y - Ag \|_{\ell_1} \] provided that the support of the vector of errors is not too large, \(\|e\|_{\ell_0} := |\{i : e_i \neq 0\}| \le \rho \cdot m\) for some \(\rho > 0\). In short, \(f\) can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; \(f\) is recovered exactly even in situations where a significant fraction of the output is corrupted.
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              Bayesian Compressive Sensing

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                Author and article information

                Journal
                IEEE Transactions on Signal Processing
                IEEE Trans. Signal Process.
                Institute of Electrical and Electronics Engineers (IEEE)
                1053-587X
                1941-0476
                May 2011
                May 2011
                : 59
                : 5
                : 1985-2001
                Article
                10.1109/TSP.2011.2107904
                5c31f281-dc6f-4a07-8197-d65924076cb9
                © 2011
                History

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