@article {ghadermarzy2013ncs,
	title = {Non-convex compressed sensing using partial support information},
	journal = {Journal of Sampling Theory in Signal and Image Processing},
	volume = {13},
	number = {3},
	year = {2014},
	pages = {249-270},
	abstract = {In this paper we address the recovery conditions of weighted $\ell_p$ minimization for signal reconstruction from compressed sensing measurements when partial support in- formation is available. We show that weighted $\ell_p$ minimization with 0 < p < 1 is stable and robust under weaker sufficient conditions compared to weighted $\ell_1$ minimization. Moreover, the sufficient recovery conditions of weighted $\ell_p$ are weaker than those of regular $\ell_p$ minimization if at least 50\% of the support estimate is accurate. We also review some algorithms which exist to solve the non-convex $\ell_p$ problem and illustrate our results with numerical experiments.},
	keywords = {compressed sensing, nonconvex optimization, sparse reconstruction, weighted $\ell_p$},
	url = {http://arxiv.org/abs/1311.3773},
	author = {Navid Ghadermarzy and Hassan Mansour and Ozgur Yilmaz}
}
@article {mansour2013GEOPiwr,
	title = {Improved wavefield reconstruction from randomized sampling via weighted one-norm minimization},
	journal = {Geophysics},
	volume = {78},
	number = {5},
	year = {2013},
	month = {08},
	pages = {V193-V206},
	abstract = {Missing-trace interpolation aims to recover the gaps caused by physical obstacles or deliberate subsampling to control acquisition costs in otherwise regularly sampled seismic wavefields. Although transform-domain sparsity promotion has proven to be an effective tool to solve this recovery problem, current recovery techniques do not fully utilize a priori information derived from the locations of the transform-domain coefficients, especially when curvelet domain sparsity is exploited. We use recovery by weighted one-norm minimization, which exploits correlations between the locations of significant curvelet coefficients of different partitions, e.g., shot records, common-offset gathers, or frequency slices of the acquired data. We use these correlations to define a sequence of 2D curvelet-based recovery problems that exploit 3D continuity exhibited by seismic wavefields without relying on the highly redundant 3D curvelet transform. To test the performance of our weighted algorithm, we compared recoveries from different data sorting and partitioning scenarios for a seismic line from the Gulf of Suez. These tests demonstrated that our method is superior to standard $\ell_1$ minimization in terms of antialiasing capability, reconstruction quality and computational memory requirements.},
	keywords = {compressed sensing, randomized sampling, Trace interpolation, weighted one-norm minimization},
	doi = {10.1190/geo2012-0383.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2013/mansour2013GEOPiwr/mansour2013GEOPiwr.pdf},
	author = {Hassan Mansour and Felix J. Herrmann and Ozgur Yilmaz}
}
@conference {mansour2012SSPwspgl1,
	title = {Beyond $\ell_1$ norm minimization for sparse signal recovery},
	booktitle = {2012 IEEE Statistical Signal Processing Workshop (SSP) (SSP{\textquoteright}12)},
	year = {2012},
	month = {03},
	publisher = {IEEE},
	organization = {IEEE},
	address = {Ann Arbor, Michigan, USA},
	abstract = {Sparse signal recovery has been dominated by the basis pursuit denoise (BPDN) problem formulation for over a decade. In this paper, we propose an algorithm that outperforms BPDN in finding sparse solutions to underdetermined linear systems of equations at no additional computational cost. Our algorithm, called WSPGL1, is a modification of the spectral projected gradient for $\ell_1$ minimization (SPGL1) algorithm in which the sequence of LASSO subproblems are replaced by a sequence of weighted LASSO subproblems with constant weights applied to a support estimate. The support estimate is derived from the data and is updated at every iteration. The algorithm also modifies the Pareto curve at every iteration to reflect the new weighted $\ell_1$ minimization problem that is being solved. We demonstrate through extensive simulations that the sparse recovery performance of our algorithm is superior to that of $\ell_1$ minimization and approaches the recovery performance of iterative re-weighted $\ell_1$ (IRWL1) minimization of Cand{\`e}s, Wakin, and Boyd. Moreover, our algorithm has the computational cost of a single BPDN problem.},
	keywords = {compressed sensing, iterative algorithms, partial support recovery, Sparse recovery, weighted $\ell_1$ minimization},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SSP/2012/mansour2012SSPwspgl1/mansour2012SSPwspgl1.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SSP/2012/mansour2012SSPwspgl1/mansour2012SSPwspgl1_pres.pdf},
	author = {Hassan Mansour}
}
