## Recovery of exact sparse representations in the presence of bounded noise (2005)

Venue: | IEEE Trans. on I.T |

Citations: | 80 - 5 self |

### BibTeX

@ARTICLE{Fuchs05recoveryof,

author = {Jean-jacques Fuchs},

title = {Recovery of exact sparse representations in the presence of bounded noise},

journal = {IEEE Trans. on I.T},

year = {2005},

pages = {3601--3608}

}

### Years of Citing Articles

### OpenURL

### Abstract

The purpose of this contribution is to extend some recent results on sparse representations of signals in redundant bases developed in the noise-free case to the case of noisy observations. The type of questions addressed so far is: given a (n,m)-matrix with and a vector, find a sufficient condition for to have an unique sparsest representation as a linear combination of the columns of. The answer is a bound on the number of nonzero entries of say, that guaranties that is the unique and sparsest solution of with. We consider the case where satisfies the sparsity conditions requested in the noise-free case and seek conditions on, a vector of additive noise or modeling errors, under which can be recovered from in a sense to be defined. 1.

### Citations

1673 | Atomic decomposition by basis pursuit - Chen, Donoho, et al. - 1998 |

1082 |
Practical Methods of Optimization
- Fletcher
- 1987
(Show Context)
Citation Context ...ditions that are also sufficient since the problem is convex. We adopt a more direct path. Since the criterion is nonsmooth at zero because of the presence . of , we introduce the sub-differential of =-=[10]-=-, a set of vectors called the subgradients, denoted : where when and when . A necessary and sufficient condition (NSC) for to be a global minimum of (QP) is that the vector zero is a sub-gradient of t... |

750 | Least angle regression - Efron, Hastie, et al. - 2004 |

527 | Greed is good: Algorithmic results for sparse approximation - Tropp - 2004 |

357 | Uncertainty principles and ideal atomic decomposition
- Donoho, Huo
- 2001
(Show Context)
Citation Context ...minimize ) . the norm of instead of the sparsity itself. The problem is then to determine sufficient conditions for the two criteria to have the same unique solution. This problem has been studied in =-=[2]-=- and later been refined in [3] in the special case where is the union of two orthonormal matrices. . "$ " % ( 4 6 + These results have been extended in [4] and [5] to matrices with an arbitrary number... |

318 |
Sparse approximate solutions to linear systems
- Natarajan
- 1995
(Show Context)
Citation Context ...has been shown that if the number of nonzero entries in is smaller than a given bound, then is the unique sparsest representation. Since searching for the sparsest representation is a NP hard problem =-=[1]-=- that can only be solved by exhaustive search, one is tempted to replace the true search for the sparsest solution : " % ( ) + "$ with the number of nonzero entries in , by the easy-to-solve linear pr... |

302 | Just relax: Convex programming methods for identifying sparse signals - Tropp |

299 | Stable recovery of sparse overcomplete representations in the presence of noise - Donoho, Elad, et al. |

215 |
Sparse representations in unions of bases
- Gribonval, Nielsen
(Show Context)
Citation Context ...e solution. This problem has been studied in [2] and later been refined in [3] in the special case where is the union of two orthonormal matrices. . "$ " % ( 4 6 + These results have been extended in =-=[4]-=- and [5] to matrices with an arbitrary number of unit norm columns. We consider here an extension of this problem. We assume the signal to have an exact sparse representation but we observe it in addi... |

183 | The Theory of Matrices in Numerical Analysis - Householder - 1964 |

174 | A generalized uncertainty principle and sparse representation in pairs of bases
- Elad, Bruckstein
(Show Context)
Citation Context ...ad of the sparsity itself. The problem is then to determine sufficient conditions for the two criteria to have the same unique solution. This problem has been studied in [2] and later been refined in =-=[3]-=- in the special case where is the union of two orthonormal matrices. . "$ " % ( 4 6 + These results have been extended in [4] and [5] to matrices with an arbitrary number of unit norm columns. We cons... |

149 | Handbook of Matrices - Lütkepohl - 1996 |

108 | Sparse nonnegative solution of underdetermined linear equations by linear programming - Donoho, Tanner - 2005 |

89 | Just relax: Convex programming methods for subset selection and sparse approximation - Tropp - 2004 |

70 | Approximation of functions over redundant dictionaries using coherence
- Gilbert, Muthukrishnan, et al.
- 2003
(Show Context)
Citation Context ... a convex program that can be transformed into an easyto-solve linear program. To state the sufficient sparsity conditions to be satisfied by , we define the coherence of the set of vectors K = to be =-=[8]-=-: = ;K N K = ; E (1) N i.e. the maximal inner product between two distinct vectors. In [2] this problem is considered when is the concatenation of two orthonormal matrices and it is shown that if : th... |

53 | On sparse representation in pairs of bases - Feuer, Nemirovski |

28 | Optimal sparse representations in general overcomplete bases - Malioutov, Cetin, et al. |

24 | On the application of the global matched filter to DOA estimation with uniform circular arrays
- Fuchs
- 2001
(Show Context)
Citation Context ...P) is in turn identical to : * . + * , + . (1 6 ( ; N ; . " 4 N F . G tiontermin(QP):* with . The two programs (QP) and (DQP) can be transformed into quadratic programs that are equivalent by duality =-=[9]-=-. In the sequel, we mainly concentrate on (QP) which is the most convenient for our purpose. It is worth noting the analogy between (QP) and the combination of (3) and (4), the regularizareplacing in ... |

22 | More on sparse representations in arbitrary bases
- Fuchs
- 2004
(Show Context)
Citation Context ...on. This problem has been studied in [2] and later been refined in [3] in the special case where is the union of two orthonormal matrices. . "$ " % ( 4 6 + These results have been extended in [4] and =-=[5]-=- to matrices with an arbitrary number of unit norm columns. We consider here an extension of this problem. We assume the signal to have an exact sparse representation but we observe it in additive noi... |

20 | Extension of the Pisarenko method to sparse linear arrays - Fuchs - 1997 |

19 | Grassmanian frames with applications to coding and communication - Strohmer, Heath - 2003 |

17 | Multipath time-delay detection and estimation - Fuchs - 1999 |

14 | Detection and estimation of superimposed signals - Fuchs - 1998 |

11 |
Nonregular regression
- Smith
- 1994
(Show Context)
Citation Context ...In the linear regression context, this is known as the subset selection or selection of variables problem and is indeed a difficult detection problem for which only ad hoc solution have been proposed =-=[7]-=-. This paper is organized as follows. The known results are described in section 2, the problem is formalized in section 3 where an mainly unrealistic (because too time consuming) maximum likelihood a... |

4 | the application of the global matched filter to DOA estimation with uniform circular arrays - “On - 2001 |

3 | Minimum L -norm reconstruction function for oversampled signals: Applications to time-delay estimation - Fuchs, Delyon - 2000 |

2 | On the Stability of Basis Pursuit in the Presence of Noise, unpublished manuscript - Donoho, Elad - 2004 |