Nonlinear compressed sensing based on composite mappings and its pointwise linearization

Classical compressed sensing (CS) allows us to recover structured signals from far few linear measurements than traditionally prescribed, thereby efficiently decreasing sampling rates. However, if there exist nonlinearities in the measurements, is it still possible to recover sparse or structured signals from the nonlinear measurements? The research of nonlinear CS is devoted to answering this question. In this paper, unlike the existing research angles of nonlinear CS, we study it from the perspective of mapping decomposition, and propose a new concept, namely, nonlinear CS based on composite mappings. Through the analysis of two forms of a nonlinear composite mapping Phi, i.e., Phi(x) = F(Ax) and Phi(x) = AF(x), we give the requirements respectively for the sensing matrix A and the nonlinear mapping F when reconstructing all sparse signals exactly from the nonlinear measurements Phi(x). Besides, we also provide a special pointwise linearization method, which can turn the nonlinear composite mapping Phi, at each point in its domain, into an equivalent linear composite mapping. This linearization method can guarantee the exact recovery of all given sparse signals even if Phi is not an injection for all sparse signals. It may help us build an algorithm framework for the composite nonlinear CS in which we can take full advantage of the existing recovery algorithms belonging to linear CS.

[1]  R. DeVore,et al.  Compressed sensing and best k-term approximation , 2008 .

[2]  S. Sastry,et al.  Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming , 2011, 1111.6323.

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

[4]  Thomas Blumensath,et al.  Compressed Sensing With Nonlinear Observations and Related Nonlinear Optimization Problems , 2012, IEEE Transactions on Information Theory.

[5]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[6]  J R Fienup,et al.  Phase retrieval algorithms: a comparison. , 1982, Applied optics.

[7]  Richard G. Baraniuk,et al.  1-Bit compressive sensing , 2008, 2008 42nd Annual Conference on Information Sciences and Systems.

[8]  Thomas Strohmer,et al.  Measure What Should be Measured: Progress and Challenges in Compressive Sensing , 2012, ArXiv.

[9]  Balas K. Natarajan,et al.  Sparse Approximate Solutions to Linear Systems , 1995, SIAM J. Comput..

[10]  Allen Y. Yang,et al.  CPRL -- An Extension of Compressive Sensing to the Phase Retrieval Problem , 2012, NIPS.

[11]  Brigitte Maier.  Handbook Of Modern Sensors Physics Designs And Applications , 2016 .

[12]  Yonina C. Eldar,et al.  Sparsity Constrained Nonlinear Optimization: Optimality Conditions and Algorithms , 2012, SIAM J. Optim..

[13]  Richard G. Baraniuk,et al.  Compressive phase retrieval , 2007, SPIE Optical Engineering + Applications.

[14]  Allen Y. Yang,et al.  Nonlinear basis pursuit , 2013, 2013 Asilomar Conference on Signals, Systems and Computers.

[15]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[16]  J R Fienup,et al.  Reconstruction of an object from the modulus of its Fourier transform. , 1978, Optics letters.

[17]  Yaniv Plan,et al.  One‐Bit Compressed Sensing by Linear Programming , 2011, ArXiv.

[18]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[19]  Guanzheng Tan,et al.  Invariance of the spark, NSP order and RIP order under elementary transformations of matrices , 2015, ArXiv.

[20]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[21]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[22]  Laurent Jacques,et al.  Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors , 2011, IEEE Transactions on Information Theory.

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