#### DMCA

## Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements (2006)

### Cached

### Download Links

Venue: | CISS 2006 (40th Annual Conference on Information Sciences and Systems |

Citations: | 108 - 7 self |

### Citations

2716 | Atomic Decomposition by Basis Pursuit
- Donoho, Saunders
- 1968
(Show Context)
Citation Context ...arting from the seminal paper [15] (see Theorem 8, attributed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation =-=[8]-=-, [14], [16], [17], [13], [19], [24], [25], [26], [11], [9], [10], [12], [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show th... |

2620 | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information
- Candès, Romberg, et al.
(Show Context)
Citation Context ..., and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], [24], [25], [26], [11], [9], [10], [12], =-=[2]-=-, [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Vershynin Department of Mathematics Univer... |

1505 | Near-optimal signal recovery from random projections: universal encoding strategies - Candés, Tao |

1398 | Decoding by linear programming
- Candès, Tao
(Show Context)
Citation Context ...). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], [24], [25], [26], [11], [9], [10], [12], [2], [1], [4], =-=[5]-=-, [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Vershynin Department of Mathematics University of Califor... |

911 | Greed is good: Algorithmic results for sparse approximation
- Tropp
(Show Context)
Citation Context ...heorem 8, attributed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], [24], =-=[25]-=-, [26], [11], [9], [10], [12], [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Vershyni... |

876 | The Dantzig selector: Statistical estimation when p is much larger than n - Candès, Tao |

629 | Optimally sparse representation in general (nonorthogonal) dictionaries via ` minimization
- Donoho, Elad
- 2003
(Show Context)
Citation Context ... paper [15] (see Theorem 8, attributed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], =-=[13]-=-, [19], [24], [25], [26], [11], [9], [10], [12], [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximat... |

579 | Uncertainty principles and ideal atomic decompositions
- Donoho, Huo
- 2001
(Show Context)
Citation Context ...g from the seminal paper [15] (see Theorem 8, attributed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], =-=[14]-=-, [16], [17], [13], [19], [24], [25], [26], [11], [9], [10], [12], [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the... |

567 | For most large underdetermined systems of linear equations the minimal ℓ1 solution is also the sparsest solution
- Donoho
(Show Context)
Citation Context ...ed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], [24], [25], [26], [11], =-=[9]-=-, [10], [12], [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Vershynin Department of M... |

436 |
The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts
- Pisier
- 1989
(Show Context)
Citation Context ...onsidered as a set in C n is at most K with respect to the norm � · �X (this was stated as the last containment in (11)). It follows that N(B r 1, � · �, u) ≤ (1 + 2K/u) r for all r > 0, see (5.7) in =-=[Pi]-=-. The set D r,n 1 consists of d(r, n) = �r � T j=1 balls of form B1 , thus � n i N � D r,n 1 , � · �X, u � ≤ d(n, r)(1 + 2K/u) r . (13) Now we combine the estimate of the covering number N(u) = log 1/... |

336 |
Sparse representations in unions of bases
- Gribonval, Nielsen
- 2003
(Show Context)
Citation Context ... [15] (see Theorem 8, attributed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], =-=[19]-=-, [24], [25], [26], [11], [9], [10], [12], [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Ro... |

269 |
Uncertainty principles and signal recovery
- Donoho, Stark
- 1989
(Show Context)
Citation Context ...he idea of convex relaxation became truly promising. It was put forward most enthusiastically and successfully by Donoho and his collaborators since the late eighties, starting from the seminal paper =-=[15]-=- (see Theorem 8, attributed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19],... |

246 | A generalized uncertainty principle and sparse representation in pairs of bases
- Elad, Bruckstein
- 2002
(Show Context)
Citation Context ... the seminal paper [15] (see Theorem 8, attributed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], =-=[16]-=-, [17], [13], [19], [24], [25], [26], [11], [9], [10], [12], [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the spars... |

181 | Quantitative robust uncertainty principles and optimally sparse decompositions
- Candes, Romberg
- 2006
(Show Context)
Citation Context ... Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], [24], [25], [26], [11], [9], [10], [12], [2], =-=[1]-=-, [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Vershynin Department of Mathematics University ... |

170 |
Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of noise
- Donoho, Elad, et al.
- 2006
(Show Context)
Citation Context ...tributed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], [24], [25], [26], =-=[11]-=-, [9], [10], [12], [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Vershynin Department... |

161 | Random vectors in the isotropic position
- Rudelson
- 1999
(Show Context)
Citation Context ...� � ≤ ε (8) provided k satisfies (7) (with constant C that may depend on K). Theorem 3.4 is proved by the techniques developed in Probability in Banach spaces. The general roadmap is similar to [21], =-=[22]-=-. We first observe that E 1 k � i∈Ω x T i ⊗ xT i = 1 n n� i=1 x T i ⊗ xT i = id C n, so the random operator whose norm we estimate in (8) has mean zero. Then the standard symmetrization (see [27] Lemm... |

120 | Geometric approach to error correcting codes and reconstruction of signals
- Rudelson, Vershynin
- 2005
(Show Context)
Citation Context ...ere is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], [24], [25], [26], [11], [9], [10], [12], [2], [1], [4], [5], =-=[23]-=-, [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Vershynin Department of Mathematics University of California, D... |

107 |
On Milman’s inequality and random subspaces which escape through a mesh in Rn.
- Gordon
- 1988
(Show Context)
Citation Context ...bspace does not intersect D. However, to obtain good constants as in (15), we will need to (a) improve the constant of embedding into D from [20], and (b) use Gordon’s Escape Through the Mesh Theorem =-=[18]-=-, which is tight in terms of constants. In Gordon’s theorem, one measures the size of a set S in R n by its Gaussian width w(D) = E sup〈g, x〉, x∈S where g is a random vector in R n whose components ar... |

103 | Just relax : Convex programming methods for subset selection and sparse approximation,”
- Tropp
- 2006
(Show Context)
Citation Context ...all available methods of Linear Programming. Convex relaxation of sparse recovery problems can be traced back in its rudimentary form to mid-seventies; references to its early history can be found in =-=[26]-=-. With the development of fast methods of Linear Programming in the eighties, the idea of convex relaxation became truly promising. It was put forward most enthusiastically and successfully by Donoho ... |

81 | On sparse representation in pairs of bases
- Feuer, Nemirovski
- 2003
(Show Context)
Citation Context ...eminal paper [15] (see Theorem 8, attributed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], =-=[17]-=-, [13], [19], [24], [25], [26], [11], [9], [10], [12], [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse appr... |

46 |
Recovery of short, complex linear combinations via l1 minimization
- Tropp
- 2005
(Show Context)
Citation Context ...(see Theorem 8, attributed there to Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], =-=[24]-=-, [25], [26], [11], [9], [10], [12], [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Ve... |

33 | Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces. - Carl - 1985 |

5 |
Compresed Sensing
- Donoho
- 2006
(Show Context)
Citation Context ... Logan, and Theorem 9). There is extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], [24], [25], [26], [11], [9], [10], =-=[12]-=-, [2], [1], [4], [5], [23], [3], [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Vershynin Department of Mathematics U... |

4 |
Error correction via
- Candes, Rudelson, et al.
(Show Context)
Citation Context ... extensive work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], [24], [25], [26], [11], [9], [10], [12], [2], [1], [4], [5], [23], =-=[3]-=-, [6], [20]. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Vershynin Department of Mathematics University of California, Davis ... |

3 |
Reconstruction and subgaussian operators, preprint
- Mendelson, Pajor, et al.
(Show Context)
Citation Context ... work being carried out, both in theory and in practice, based on the convex relaxation [8], [14], [16], [17], [13], [19], [24], [25], [26], [11], [9], [10], [12], [2], [1], [4], [5], [23], [3], [6], =-=[20]-=-. To have theoretical guarantees for the convex relaxation method, one needs to show that the sparse approximation Roman Vershynin Department of Mathematics University of California, Davis Davis, Cali... |

3 |
Approximate John’s decompositions, Operator Theory
- Rudelson
(Show Context)
Citation Context ...T i � � � ≤ ε (8) provided k satisfies (7) (with constant C that may depend on K). Theorem 3.4 is proved by the techniques developed in Probability in Banach spaces. The general roadmap is similar to =-=[21]-=-, [22]. We first observe that E 1 k � i∈Ω x T i ⊗ xT i = 1 n n� i=1 x T i ⊗ xT i = id C n, so the random operator whose norm we estimate in (8) has mean zero. Then the standard symmetrization (see [27... |