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Iteratively reweighted least squares minimization for sparse recovery. (English) Zbl 1202.65046
From the authors’ abstract: “Under certain conditions (known as the restricted isometry property, or RIP) on the $$m \times N$$ matrix $$\Phi$$ (where $$m < N$$), vectors $$x \in \mathbb R^N$$ that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from $$y := \Phi x$$ even though $$\Phi ^{-1}(y)$$ is typically an $$(N - m)$$-dimensional hyperplane; in addition, $$x$$ is then equal to the element in $$\Phi ^{-1}(y)$$ of minimal $$\ell _1$$-norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining $$x$$, as the limit of an iteratively reweighted least squares algorithm. We prove that when $$\Phi$$ satisfies the RIP conditions, the sequence $$x^{(n)}$$ converges for all $$y$$, regardless of whether $$\Phi ^{-1}(y)$$ contains a sparse vector.”

##### MSC:
 65F20 Numerical solutions to overdetermined systems, pseudoinverses 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 65K10 Numerical optimization and variational techniques 90C30 Nonlinear programming
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