## A remark on compressed sensing (2007)

### Cached

### Download Links

Citations: | 21 - 0 self |

### BibTeX

@TECHREPORT{Kashin07aremark,

author = {B. S. Kashin and V. N. Temlyakov},

title = {A remark on compressed sensing},

institution = {},

year = {2007}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract—Recently, a new direction in signal processing – “Compressed Sensing " is being actively developed. A number of authors have pointed out a connection between the Compressed Sensing problem and the problem of estimating the Kolmogorov widths, studied in the seventies and eighties of the last century. In this paper we make the above mentioned connection more precise. DOI: 10.1134/S0001434607110193

### Citations

1885 | Compressed sensing
- Donoho
- 2006
(Show Context)
Citation Context ...e surveys [C], [D]) in Compressed Sensing resulted in proving the existence of matrices Φ with k(m, n) ≍ n/ log(m/n) which is substantially larger than n 1/2 . A number of authors (see, for instance, =-=[Do]-=-, [CDD]) have pointed out a connection between the Compressed Sensing problem and the problem of estimating the widths of finite dimensional sets, studied at the end of seventies and the beginning of ... |

1801 | Atomic Decomposition by Basis Pursuit
- Chen, Donoho, et al.
- 1998
(Show Context)
Citation Context ...ery problem can be stated as the problem of finding the sparsest vector u 0 := u 0 Φ (y) ∈ Rm : (P0) min �v�0 subject to Φv = y, where �v�0 := | supp(v)|. D. Donoho with coauthors (see, for instance, =-=[CDS]-=- and [DET] and history therein) have suggested an economical algorithm and have begun a systematic 1sstudy of the following question. For which measurement matrices Φ the highly non-convex combinatori... |

810 |
Stable signal recovery from incomplete and inaccurate measurements
- Candes, Romberg, et al.
- 2005
(Show Context)
Citation Context ...lso proved existence of sensing matrices Φ obeying the condition δ2S +δ3S < 1 for large values of sparsity S ≍ n/ log(m/n). For a positive number a denote σa(v)1 := min w∈Rm �v − w�1. :|supp(w)|≤a In =-=[CRT]-=- the authors proved that if δ3S + 3δ4S < 2, then (1.6) �u − AΦ(Φu)�2 ≤ CS −1/2 σS(u)1. We note that properties of the RIP-type matrices have already been imployed in [K] for the widths estimation. The... |

704 | Decoding by linear programming
- Candes, Tao
- 2005
(Show Context)
Citation Context ...log m n )1/2 , (A2) then, for any u ∈ Bm1 , we have ‖u−AΦ(Φu)‖2 ≤ C ( log m n )1/2 . We now proceed to the contribution of E. Candes, J. Romberg, and T. Tao published in a series of papers. They (see =-=[10]-=-) introduced the following Restricted Isometry Property (RIP) of a sensing matrix Φ: δS < 1 is the S-restricted isometry constant of Φ if it is the smallest quantity such that (1− δS)‖c‖22 ≤ ‖ΦT c‖22 ... |

701 |
An introduction to compressive sampling
- Candes, Wakin
- 2008
(Show Context)
Citation Context ...rse with k < (1 + 1/M)/2. This allows us to build rather simple deterministic matrices Φ with k(m, n) ≍ n 1/2 and recover with the ℓ1-minimization algorithm AΦ from (P1). Recent progress (see surveys =-=[C]-=-, [D]) in Compressed Sensing resulted in proving the existence of matrices Φ with k(m, n) ≍ n/ log(m/n) which is substantially larger than n 1/2 . A number of authors (see, for instance, [Do], [CDD]) ... |

321 | Stable recovery of sparse overcomplete representations in the presence of noise
- Donoho, Elad, et al.
(Show Context)
Citation Context ...m can be stated as the problem of finding the sparsest vector u 0 := u 0 Φ (y) ∈ Rm : (P0) min �v�0 subject to Φv = y, where �v�0 := | supp(v)|. D. Donoho with coauthors (see, for instance, [CDS] and =-=[DET]-=- and history therein) have suggested an economical algorithm and have begun a systematic 1sstudy of the following question. For which measurement matrices Φ the highly non-convex combinatorial optimiz... |

92 |
Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise
- Donoho, Elad, et al.
(Show Context)
Citation Context ...quivalent to the following problem of finding the sparsest vector (column) u0 := u0Φ(y) ∈ Rm: min ‖v‖0 subject to Φv = y, (P0) where ‖v‖0 := | supp(v)|. Donoho with coauthors (see, for instance, [1], =-=[2]-=- and the history therein) suggested an economical algorithm and begun a systematic study of the following question. For which *E-mail: kashin@mi.ras.ru. **E-mail: temlyak@math.sc.edu. 748 A REMARK ON ... |

42 |
The widths of certain finite-dimensional sets and classes of smooth functions
- Kashin
- 1977
(Show Context)
Citation Context ...q) = d n (B m q ′ , ℓp ′), p′ := p/(p − 1). In a particular case p = 2, q = ∞ of our interest (1.1) gives (1.2) dn(B m 2 , ℓ∞) = d n (B m 1 , ℓ2). It has been established in approximation theory (see =-=[K]-=- and [GG]) that (1.3) dn(B m 2 , ℓ∞) ≤ C((1 + log(m/n))/n) 1/2 . By C we denote here and in the whole paper an absolute constant. In other words, it was proved (see (1.3) and (1.2)) that for any pair ... |

19 |
Section of finite-dimensional sets and classes of smooth functions
- Kashin
- 1977
(Show Context)
Citation Context ...lar case p = 2, q = ∞ of our interest, (1.1) gives dn(Bm2 , ∞) = d n(Bm1 , 2). (1.2) MATHEMATICAL NOTES Vol. 82 No. 6 2007 750 KASHIN, TEMLYAKOV It has been established in approximation theory (see =-=[8]-=- and [9]) that dn(Bm2 , ∞) ≤ C ( 1 + log(m/n) n )1/2 . (1.3) By C we denote an absolute constant throughout the paper. In other words, it was proved (see(1.3) and (1.2)) that, for any pair (m,n), the... |

9 | Optimal computation
- DeVore
(Show Context)
Citation Context ...ith k < (1 + 1/M)/2. This allows us to build rather simple deterministic matrices Φ with k(m, n) ≍ n 1/2 and recover with the ℓ1-minimization algorithm AΦ from (P1). Recent progress (see surveys [C], =-=[D]-=-) in Compressed Sensing resulted in proving the existence of matrices Φ with k(m, n) ≍ n/ log(m/n) which is substantially larger than n 1/2 . A number of authors (see, for instance, [Do], [CDD]) have ... |

9 |
Diameters of sets in normed linear spaces and approximation of functions with trigonometric polynomials, Uspekhi Mat
- Ismagilov
(Show Context)
Citation Context ... − n. It is well known that the Kolmogorov and the Gel’fand widths are related by the duality formula. S.M. Nikol’skii was the first to use the duality idea in approximation theory. For instance (see =-=[I]-=-), in the case of F = B m p is a unit ℓp-ball in R m and 1 ≤ q, p ≤ ∞ one has (1.1) dn(B m p , ℓq) = d n (B m q ′ , ℓp ′), p′ := p/(p − 1). In a particular case p = 2, q = ∞ of our interest (1.1) give... |

6 |
On widths of the Euclidean ball,” Dokl
- Garnaev, Gluskin
- 1984
(Show Context)
Citation Context ... (B m q ′ , ℓp ′), p′ := p/(p − 1). In a particular case p = 2, q = ∞ of our interest (1.1) gives (1.2) dn(B m 2 , ℓ∞) = d n (B m 1 , ℓ2). It has been established in approximation theory (see [K] and =-=[GG]-=-) that (1.3) dn(B m 2 , ℓ∞) ≤ C((1 + log(m/n))/n) 1/2 . By C we denote here and in the whole paper an absolute constant. In other words, it was proved (see (1.3) and (1.2)) that for any pair (m, n) th... |

3 |
Compressed Sensing and k-Term Approximation (Manuscript
- Cohen, Dahmen, et al.
- 2007
(Show Context)
Citation Context ...eys [C], [D]) in Compressed Sensing resulted in proving the existence of matrices Φ with k(m, n) ≍ n/ log(m/n) which is substantially larger than n 1/2 . A number of authors (see, for instance, [Do], =-=[CDD]-=-) have pointed out a connection between the Compressed Sensing problem and the problem of estimating the widths of finite dimensional sets, studied at the end of seventies and the beginning of eightie... |