Robust nonnegative sparse recovery and 0/1-Bernoulli measurements

We investigate recovery of nonnegative vectors from nonadaptive compressive measurements in the presence of noise of unknown power. It is known in the literature that under additional assumptions on the measurement design recovery of such vectors is possible with nonnegative least squares without any regularization. We show that uniqueness results known for the noiseless case carry over to robust guarantees in the noisy setting. We present guarantees which hold instantaneously by connecting the relation to the robust nullspace property. As an important example, we prove that an m × n random iid. 0/1-valued Bernoulli matrix with m = O(s log(n)) rows admits the robust nullspace property with high probability and meets the design requirements for nonnegative least squares recovery. Our analysis is motivated by applications in wireless network activity detection.

[1]  N. Meinshausen.  Sign-constrained least squares estimation for high-dimensional regression , 2012, 1202.0889.

[2]  Michael P. Friedlander,et al.  Probing the Pareto Frontier for Basis Pursuit Solutions , 2008, SIAM J. Sci. Comput..

[3]  Michael Elad,et al.  On the uniqueness of non-negative sparse & redundant representations , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[4]  Holger Rauhut,et al.  An Invitation to Compressive Sensing , 2013 .

[5]  V. Koltchinskii,et al.  Bounding the smallest singular value of a random matrix without concentration , 2013, 1312.3580.

[6]  Philip Schniter,et al.  An Empirical-Bayes Approach to Recovering Linearly Constrained Non-Negative Sparse Signals , 2013, IEEE Transactions on Signal Processing.

[7]  Holger Rauhut,et al.  Stable low-rank matrix recovery via null space properties , 2015, ArXiv.

[8]  R. DeVore,et al.  A Simple Proof of the Restricted Isometry Property for Random Matrices , 2008 .

[9]  D. Donoho,et al.  Sparse nonnegative solution of underdetermined linear equations by linear programming. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[10]  I. Johnstone,et al.  Maximum Entropy and the Nearly Black Object , 1992 .

[11]  Shahar Mendelson,et al.  Learning without Concentration , 2014, COLT.

[12]  Slawomir Stanczak,et al.  Block compressed sensing based distributed resource allocation for M2M communications , 2016, 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[13]  Peter Jung,et al.  Robust Nonnegative Sparse Recovery and the Nullspace Property of 0/1 Measurements , 2016, IEEE Transactions on Information Theory.

[14]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.

[15]  Holger Rauhut,et al.  On the gap between RIP-properties and sparse recovery conditions , 2015, ArXiv.

[16]  Simon Foucart,et al.  Sparse Recovery by Means of Nonnegative Least Squares , 2014, IEEE Signal Processing Letters.

[17]  Emmanuel J. Candès,et al.  PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming , 2011, ArXiv.

[18]  Matthias Hein,et al.  Non-negative least squares for high-dimensional linear models: consistency and sparse recovery without regularization , 2012, 1205.0953.

[19]  David L. Donoho,et al.  Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[20]  Matthias Hein,et al.  Sparse recovery by thresholded non-negative least squares , 2011, NIPS.

[21]  Shuheng Zhou,et al.  25th Annual Conference on Learning Theory Reconstruction from Anisotropic Random Measurements , 2022 .

[22]  S. Jiao,et al.  Compressed sensing and reconstruction with bernoulli matrices , 2010, The 2010 IEEE International Conference on Information and Automation.

[23]  Richard Kueng,et al.  RIPless compressed sensing from anisotropic measurements , 2012, ArXiv.

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

[25]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[26]  Ao Tang,et al.  A Unique “Nonnegative” Solution to an Underdetermined System: From Vectors to Matrices , 2010, IEEE Transactions on Signal Processing.