On the phase transition of corrupted sensing

In [1], a sharp phase transition has been numerically observed when a constrained convex procedure is used to solve the corrupted sensing problem. In this paper, we present a theoretical analysis for this phenomenon. Specifically, we establish the threshold below which this convex procedure fails to recover signal and corruption with high probability. Together with the work in [1], we prove that a sharp phase transition occurs around the sum of the squares of spherical Gaussian widths of two tangent cones. Numerical experiments are provided to demonstrate the correctness and sharpness of our results.

[1]  John Wright,et al.  Dense Error Correction via L1-Minimization , 2008, 0809.0199.

[2]  Trac D. Tran,et al.  Exact Recoverability From Dense Corrupted Observations via $\ell _{1}$-Minimization , 2011, IEEE Transactions on Information Theory.

[3]  Allen Y. Yang,et al.  Robust Face Recognition via Sparse Representation , 2009, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  Richard G. Baraniuk,et al.  Stable Restoration and Separation of Approximately Sparse Signals , 2011, ArXiv.

[5]  John Wright,et al.  Dense Error Correction Via $\ell^1$-Minimization , 2010, IEEE Transactions on Information Theory.

[6]  Trac D. Tran,et al.  Robust Lasso With Missing and Grossly Corrupted Observations , 2011, IEEE Transactions on Information Theory.

[7]  Joel A. Tropp,et al.  Sharp Recovery Bounds for Convex Demixing, with Applications , 2012, Found. Comput. Math..

[8]  R. Nowak,et al.  Compressed Sensing for Networked Data , 2008, IEEE Signal Processing Magazine.

[9]  Joel A. Tropp,et al.  Living on the edge: phase transitions in convex programs with random data , 2013, 1303.6672.

[10]  Stephen P. Boyd,et al.  Graph Implementations for Nonsmooth Convex Programs , 2008, Recent Advances in Learning and Control.

[11]  Helmut Bölcskei,et al.  Recovery of Sparsely Corrupted Signals , 2011, IEEE Transactions on Information Theory.

[12]  Xiaodong Li,et al.  Compressed Sensing and Matrix Completion with Constant Proportion of Corruptions , 2011, Constructive Approximation.

[13]  Ehsan Elhamifar,et al.  Sparse subspace clustering , 2009, 2009 IEEE Conference on Computer Vision and Pattern Recognition.

[14]  Yulong Liu,et al.  Corrupted sensing with sub-Gaussian measurements , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[15]  Christoph Studer,et al.  Probabilistic Recovery Guarantees for Sparsely Corrupted Signals , 2012, IEEE Transactions on Information Theory.

[16]  M. Talagrand,et al.  Probability in Banach Spaces: Isoperimetry and Processes , 1991 .

[17]  Gábor Lugosi,et al.  Concentration Inequalities - A Nonasymptotic Theory of Independence , 2013, Concentration Inequalities.

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

[19]  MaYi,et al.  Dense error correction via l1-minimization , 2010 .

[20]  Joel A. Tropp,et al.  Universality laws for randomized dimension reduction, with applications , 2015, ArXiv.

[21]  Rina Foygel,et al.  Corrupted Sensing: Novel Guarantees for Separating Structured Signals , 2013, IEEE Transactions on Information Theory.