Structure-Blind Signal Recovery

We consider the problem of recovering a signal observed in Gaussian noise. If the set of signals is convex and compact, and can be specified beforehand, one can use classical linear estimators that achieve a risk within a constant factor of the minimax risk. However, when the set is unspecified, designing an estimator that is blind to the hidden structure of the signal remains a challenging problem. We propose a new family of estimators to recover signals observed in Gaussian noise. Instead of specifying the set where the signal lives, we assume the existence of a well-performing linear estimator. Proposed estimators enjoy exact oracle inequalities and can be efficiently computed through convex optimization. We present several numerical illustrations that show the potential of the approach.

[1]  L. Wasserman.  All of Nonparametric Statistics , 2005 .

[2]  O. Lepskii.  On a Problem of Adaptive Estimation in Gaussian White Noise , 1991 .

[3]  S. Haykin,et al.  Adaptive Filter Theory , 1986 .

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

[5]  A. Juditsky,et al.  Nonparametric Denoising of Signals with Unknown Local Structure, I: Oracle Inequalities , 2008, 0809.0814.

[6]  Nora Serdyukova,et al.  Adaptive estimation under single-index constraint in a regression model , 2013, 1304.7668.

[7]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing - The Sparse Way, 3rd Edition , 2008 .

[8]  I. A. Ibragimov,et al.  Estimation of Linear Functionals in Gaussian Noise , 1988 .

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

[10]  Benjamin Recht,et al.  Atomic norm denoising with applications to line spectral estimation , 2011, Allerton.

[11]  S. Mallat.  A wavelet tour of signal processing , 1998 .

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

[13]  Ali H. Sayed,et al.  Linear Estimation (Information and System Sciences Series) , 2000 .

[14]  I. Ibragimov,et al.  On Nonparametric Estimation of the Value of a Linear Functional in Gaussian White Noise , 1985 .

[15]  A. Juditsky,et al.  Nonparametric estimation by convex programming , 2009, 0908.3108.

[16]  Yurii Nesterov,et al.  On first-order algorithms for l1/nuclear norm minimization , 2013, Acta Numerica.

[17]  Zaïd Harchaoui,et al.  Adaptive Recovery of Signals by Convex Optimization , 2015, COLT.

[18]  A. Juditsky,et al.  Functional aggregation for nonparametric regression , 2000 .

[19]  A. Juditsky,et al.  Nonparametric denoising signals of unknown local structure, II: Nonparametric function recovery , 2010 .

[20]  Alexandre B. Tsybakov,et al.  Introduction to Nonparametric Estimation , 2008, Springer series in statistics.

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

[22]  P. Massart,et al.  Adaptive estimation of a quadratic functional by model selection , 2000 .

[23]  A. Juditsky,et al.  On detecting harmonic oscillations , 2013, 1301.5328.

[24]  D. Donoho,et al.  Renormalization Exponents and Optimal Pointwise Rates of Convergence , 1992 .

[25]  D. Donoho.  Statistical Estimation and Optimal Recovery , 1994 .