Recovery of sign vectors in quadratic compressed sensing

In certain applications, recovering the signs of values may be more critical than the values themselves. Inspired by advances of sparse recovery of signals with fewer measurements, we would like to study the sign recovery problem and generalize it from a linear case to a non-linear setup. We focus on the sign values in quadratic measurement systems and provide theorems for the consistency condition, which ensures the signs are recovered correctly with probability close to 1. In deriving the consistency condition, we adopt a new penalty term using the trace operation and transform the optimization problem to the widely known Lasso problem. We also present simulation results to verify the correctness of our theorems.

[1]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[2]  Eckart Meese,et al.  High-throughput miRNA profiling of human melanoma blood samples , 2010, BMC Cancer.

[3]  Bin Yu,et al.  On Model Selection Consistency of the Elastic Net When p >> n , 2008 .

[4]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[5]  Pablo A. Parrilo,et al.  The Convex Geometry of Linear Inverse Problems , 2010, Foundations of Computational Mathematics.

[6]  Alexandros G. Dimakis,et al.  Sparse Recovery of Nonnegative Signals With Minimal Expansion , 2011, IEEE Transactions on Signal Processing.

[7]  Stephen P. Boyd,et al.  Proximal Algorithms , 2013, Found. Trends Optim..

[8]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[9]  Olgica Milenkovic,et al.  Subspace Pursuit for Compressive Sensing Signal Reconstruction , 2008, IEEE Transactions on Information Theory.

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

[11]  Stefanie S Jeffrey,et al.  Cancer biomarker profiling with microRNAs , 2008, Nature Biotechnology.

[12]  Michel Verhaegen,et al.  Quadratic Basis Pursuit , 2013, 1301.7002.

[13]  Gitta Kutyniok,et al.  Theory and applications of compressed sensing , 2012, 1203.3815.

[14]  Peng Zhao,et al.  On Model Selection Consistency of Lasso , 2006, J. Mach. Learn. Res..

[15]  Martin J. Wainwright,et al.  Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using $\ell _{1}$ -Constrained Quadratic Programming (Lasso) , 2009, IEEE Transactions on Information Theory.

[16]  Jun Zou,et al.  Micro-RNA profile detection via factor graphs , 2016, 2016 IEEE 17th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC).