@article {saab2008ACHAsrb,
	title = {Sparse recovery by non-convex optimization - instance optimality},
	journal = {Applied and Computational Harmonic Analysis},
	volume = {29},
	number = {1},
	year = {2010},
	month = {07},
	pages = {30-48},
	abstract = {In this note, we address the theoretical properties of $\Delta_p$, a class of compressed sensing decoders that rely on $l^p$ minimization with $p \in (0,1)$ to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. In particular, we extend the results of Cand{\textquoteleft}es, Romberg and Tao [3] and Wojtaszczyk [30] regarding the decoder $\Delta_1$, based on $\ell^1$ minimization, to $Δp$ with $p \in (0,1)$. Our results are two-fold. First, we show that under certain sufficient conditions that are weaker than the analogous sufficient conditions for $\Delta_1$ the decoders $\Delta_p$ are robust to noise and stable in the sense that they are $(2,p)$ instance optimal. Second, we extend the results of Wojtaszczyk to show that, like $\Delta_1$, the decoders $\Delta_p$ are (2,2) instance optimal in probability provided the measurement matrix is drawn from an appropriate distribution. While the extension of the results of [3] to the setting where $p \in (0,1)$ is straightforward, the extension of the instance optimality in probability result of [30] is non-trivial. In particular, we need to prove that the $LQ_1$ property, introduced in [30], and shown to hold for Gaussian matrices and matrices whose columns are drawn uniformly from the sphere, generalizes to an $LQ_p$ property for the same classes of matrices. Our proof is based on a result by Gordon and Kalton [18] about the Banach-Mazur distances of p-convex bodies to their convex hulls.},
	keywords = {Compressive Sensing, non-convex},
	doi = {10.1016/j.acha.2009.08.002},
	url = {http://www.sciencedirect.com/science/article/pii/S1063520309000864},
	url2 = {https://slim.gatech.edu/Publications/Public/Journals/ACHA/2010/saab2008ACHAsrb/saab2008ACHAsrb.pdf},
	author = {Rayan Saab and Ozgur Yilmaz}
}
@conference {saab2009SAMPTAnccs,
	title = {A short note on non-convex compressed sensing},
	booktitle = {SAMPTA technical program},
	year = {2009},
	publisher = {SAMPTA},
	organization = {SAMPTA},
	abstract = {In this note, we summarize the results we recently proved in\citeSY08 on the theoretical performance guarantees of the decoders $\Delta_p$. These decoders rely on $\ell^p$ minimization with $p {\i}n (0,1)$ to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. Our guarantees generalize the results of \citeCRT05 and \citeWojtaszczyk08 about decoding by $\ell_p$ minimization with $p = 1$, to the setting where $p {\i}n (0,1)$ and are obtained under weaker sufficient conditions. We also present novel extensions of our results in \citeSY08 that follow from the recent work of DeVore et al. in \citeDPW08. Finally, we show some insightful numerical experiments displaying the trade-off in the choice of $p \in (0,1]$ depending on certain properties of the input signal.},
	keywords = {Presentation},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SAMPTA/2009/saab09SAMPTAnccs/saab09SAMPTAnccs.pdf},
	author = {Rayan Saab and Ozgur Yilmaz}
}
@article {vandenberg2009ACMstf,
	title = {Sparco: a testing framework for sparse reconstruction},
	journal = {ACM Transactions on Mathematical Software},
	volume = {35},
	number = {4},
	year = {2009},
	month = {02},
	pages = {1-16},
	abstract = {Sparco is a framework for testing and benchmarking algorithms for sparse reconstruction. It includes a large collection of sparse reconstruction problems drawn from the imaging, compressed sensing, and geophysics literature. Sparco is also a framework for implementing new test problems and can be used as a tool for reproducible research. Sparco is implemented entirely in Matlab, and is released as open-source software under the GNU Public License.},
	keywords = {compressed sensing, linear operators, Sparse recovery},
	url = {http://doi.acm.org/10.1145/1462173.1462178},
	author = {Ewout van den Berg and Michael P. Friedlander and Gilles Hennenfent and Felix J. Herrmann and Rayan Saab and Ozgur Yilmaz}
}
@conference {saab2008ICASSPssa,
	title = {Stable sparse approximations via nonconvex optimization},
	booktitle = {ICASSP},
	year = {2008},
	publisher = {ICASSP},
	organization = {ICASSP},
	abstract = {We present theoretical results pertaining to the ability of lp minimization to recover sparse and compressible signals from incomplete and noisy measurements. In particular, we extend the results of Cande{\textquoteleft}s, Romberg and Tao [1] to the p < 1 case. Our results indicate that depending on the restricted isometry constants (see, e.g.,[2] and [3]) and the noise level, lp minimization with certain values of p < 1 provides better theoretical guarantees in terms of stability and robustness than l1 minimization does. This is especially true when the restricted isometry constants are relatively large.},
	keywords = {ICASSP},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/ICASSP/2008/saab08ICASSPssa/saab08ICASSPssa.pdf},
	author = {Rayan Saab and Rick Chartrand and Ozgur Yilmaz}
}
