## Temlyakov, A remark on compressed sensing (2007)

Citations: | 20 - 0 self |

### BibTeX

@TECHREPORT{Kashin07temlyakov,a,

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

title = {Temlyakov, A remark on compressed sensing},

institution = {},

year = {2007}

}

### OpenURL

### Abstract

Recently, Compressed Sensing (Compressive Sampling) has attracted a lot of attention of both mathematicians and computer scientists. Compressed Sensing refers to a problem of economical recovery of an unknown vector u ∈ R m from the information provided by linear measurements 〈u, ϕj〉, ϕj ∈ R m, j = 1,..., n. The goal is to design an algorithm

### Citations

1716 | Compressed sensing
- Donoho
- 2005
(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 ... |

1652 | Atomic decomposition by basis pursuit
- Chen, Donoho, et al.
- 2001
(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... |

742 |
Stable signal recovery from incomplete and inaccurate measurements
- Candes, Romberg, et al.
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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... |

655 | Decoding by linear programming - Candes, Tao - 2005 |

634 |
An introduction to compressive sampling
- Candès, 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]) ... |

297 | Stable recovery of sparse overcomplete representations in the presence of noise
- Donoho, Elad, et al.
- 2006
(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... |

38 |
Diameters of certain finite-dimensional sets in 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 ... |

9 | Optimal computation
- DeVore
- 2006
(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 ... |

6 |
The widths of a 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... |

6 |
Diameter of sets in normed linear spaces and the approximation of functions by trigonometric polynomials
- Ismagilov
- 1974
(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... |

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... |