Breaking the ℓ1 recovery thresholds with reweighted ℓ1 optimization

It is now well understood that ℓ1 minimization algorithm is able to recover sparse signals from incomplete measurements [2], [1], [3] and sharp recoverable sparsity thresholds have also been obtained for the ℓ1 minimization algorithm. In this paper, we investigate a new iterative reweighted ℓ1 minimization algorithm and showed that the new algorithm can increase the sparsity recovery threshold of ℓ1 minimization when decoding signals from relevant distributions. Interestingly, we observed that the recovery threshold performance of the new algorithm depends on the behavior, more specifically the derivatives, of the signal amplitude probability distribution at the origin.

[1]  B. Grünbaum.  Grassmann angles of convex polytopes , 1968 .

[2]  L. Santaló.  Geometría integral en espacios de curvatura constante , 1952 .

[3]  Weiyu Xu,et al.  On sharp performance bounds for robust sparse signal recoveries , 2009, 2009 IEEE International Symposium on Information Theory.

[4]  E.J. Candes.  Compressive Sampling , 2022 .

[5]  Weiyu Xu,et al.  Weighted ℓ1 minimization for sparse recovery with prior information , 2009, 2009 IEEE International Symposium on Information Theory.

[6]  J. K. Böröczky,et al.  Random projections of regular polytopes , 1999 .

[7]  David L. Donoho,et al.  High-Dimensional Centrally Symmetric Polytopes with Neighborliness Proportional to Dimension , 2006, Discret. Comput. Geom..

[8]  Rolf Schneider,et al.  Random projections of regular simplices , 1992, Discret. Comput. Geom..

[9]  P Mcmullen,et al.  Non-linear angle-sum relations for polyhedral cones and polytopes , 1975, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  D. Donoho,et al.  Thresholds for the Recovery of Sparse Solutions via L1 Minimization , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

[11]  B. Hassibi,et al.  Compressed sensing over the Grassmann manifold: A unified analytical framework , 2008, 2008 46th Annual Allerton Conference on Communication, Control, and Computing.

[12]  Stephen P. Boyd,et al.  Enhancing Sparsity by Reweighted ℓ1 Minimization , 2007, 0711.1612.

[13]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.