Sparse Recovery by Non-convex Optimization -- Instance Optimality

Rayan Saab; Ozgur Yilmaz

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arXiv · 0809.0745

Sparse Recovery by Non-convex Optimization -- Instance Optimality

Rayan Saab·Ozgur Yilmaz

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Abstract

In this note, we address the theoretical properties of $Δ_p$, a class of compressed sensing decoders that rely on $\ell^p$ minimization with 0<p<1 to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. In particular, we extend the results of Candes, Romberg and Tao, and Wojtaszczyk regarding the decoder $Δ_1$, based on $\ell^1$ minimization, to $Δ_p$ with 0<p<1. Our results are two-fold. First, we show that under certain sufficient conditions that are weaker than the analogous sufficient conditions for $Δ_1$ the decoders $Δ_p$ are robust to noise and stable in the sense that they are (2,p) instance optimal for a large class of encoders. Second, we extend the results of Wojtaszczyk to show that, like $Δ_1$, the decoders $Δ_p$ are (2,2) instance optimal in probability provided the measurement matrix is drawn from an appropriate distribution.

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Rayan Saab, Ozgur Yilmaz. 2009-08-10. Sparse Recovery by Non-convex Optimization -- Instance Optimality. https://arxiv.org/abs/0809.0745

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