Compressed sensing under weak moment assumptions

We prove that iid random vectors that satisfy a rather weak moment assumption can be used as measurement vectors in Compressed Sensing, and the number of measurements required for exact reconstruction is the same as the best possible estimate -- exhibited by a random gaussian matrix. We also prove that this moment condition is necessary, up to a $\log \log $ factor. Applications to the Compatibility Condition and the Restricted Eigenvalue Condition in the noisy setup and to properties of neighbourly random polytopes are also discussed.

[1]  J. Claerbout,et al.  Robust Modeling With Erratic Data , 1973 .

[2]  R. Dudley.  Central Limit Theorems for Empirical Measures , 1978 .

[3]  J. Kuelbs.  Probability on Banach spaces , 1978 .

[4]  H. L. Taylor,et al.  Deconvolution with the l 1 norm , 1979 .

[5]  F. Santosa,et al.  Linear inversion of ban limit reflection seismograms , 1986 .

[6]  K. Ball.  Cube slicing in ⁿ , 1986 .

[7]  B. Logan,et al.  Signal recovery and the large sieve , 1992 .

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

[9]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

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

[11]  R. Latala.  Estimation of moments of sums of independent real random variables , 1997 .

[12]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[13]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[14]  P. Massart,et al.  Gaussian model selection , 2001 .

[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]  R. Tibshirani,et al.  Least angle regression , 2004, math/0406456.

[17]  B. Efron.  The Estimation of Prediction Error , 2004 .

[18]  David L. Donoho,et al.  Neighborly Polytopes And Sparse Solution Of Underdetermined Linear Equations , 2005 .

[19]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[20]  S. Mendelson,et al.  Uniform Uncertainty Principle for Bernoulli and Subgaussian Ensembles , 2006, math/0608665.

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

[22]  V. Koltchinskii.  Local Rademacher complexities and oracle inequalities in risk minimization , 2006, 0708.0083.

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

[24]  Terence Tao,et al.  The Dantzig selector: Statistical estimation when P is much larger than n , 2005, math/0506081.

[25]  P. Massart,et al.  Concentration inequalities and model selection , 2007 .

[26]  S. Mendelson,et al.  Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis , 2007 .

[27]  M. Rudelson,et al.  On sparse reconstruction from Fourier and Gaussian measurements , 2008 .

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

[29]  R. Adamczak,et al.  Restricted Isometry Property of Matrices with Independent Columns and Neighborly Polytopes by Random Sampling , 2009, 0904.4723.

[30]  P. Bickel,et al.  SIMULTANEOUS ANALYSIS OF LASSO AND DANTZIG SELECTOR , 2008, 0801.1095.

[31]  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.

[32]  Martin J. Wainwright,et al.  Restricted Eigenvalue Properties for Correlated Gaussian Designs , 2010, J. Mach. Learn. Res..

[33]  V. Koltchinskii,et al.  Oracle inequalities in empirical risk minimization and sparse recovery problems , 2011 .

[34]  Sara van de Geer,et al.  Statistics for High-Dimensional Data , 2011 .

[35]  Djalil Chafaï,et al.  Interactions between compressed sensing, random matrices, and high dimensional geometry , 2012 .

[36]  S. Geer.  Weakly decomposable regularization penalties and structured sparsity , 2012, 1204.4813.

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

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

[39]  Roberto Imbuzeiro Oliveira,et al.  The lower tail of random quadratic forms with applications to ordinary least squares , 2013, ArXiv.

[40]  S. Mendelson.  A remark on the diameter of random sections of convex bodies , 2013, 1312.3608.

[41]  R. Vershynin,et al.  Covariance estimation for distributions with 2+ε moments , 2011, 1106.2775.

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

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

[44]  S. Foucart.  Stability and robustness of ℓ1-minimizations with Weibull matrices and redundant dictionaries , 2014 .

[45]  S. Geer,et al.  On higher order isotropy conditions and lower bounds for sparse quadratic forms , 2014, 1405.5995.

[46]  E. Milman,et al.  Remarks on the KLS conjecture and Hardy-type inequalities , 2014, 1405.0617.

[47]  M. Rudelson,et al.  Small Ball Probabilities for Linear Images of High-Dimensional Distributions , 2014, 1402.4492.

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

[49]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.