## One sketch for all: Fast algorithms for compressed sensing (2007)

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Venue: | In Proc. 39th ACM Symp. Theory of Computing |

Citations: | 61 - 11 self |

### BibTeX

@INPROCEEDINGS{Gilbert07onesketch,

author = {A. C. Gilbert and M. J. Strauss and R. Vershynin},

title = {One sketch for all: Fast algorithms for compressed sensing},

booktitle = {In Proc. 39th ACM Symp. Theory of Computing},

year = {2007}

}

### Years of Citing Articles

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### Abstract

Compressed Sensing is a new paradigm for acquiring the compressible signals that arise in many applications. These signals can be approximated using an amount of information much smaller than the nominal dimension of the signal. Traditional approaches acquire the entire signal and process it to extract the information. The new approach acquires a small number of nonadaptive linear measurements of the signal and uses sophisticated algorithms to determine its information content. Emerging technologies can compute these general linear measurements of a signal at unit cost per measurement. This paper exhibits a randomized measurement ensemble and a signal reconstruction algorithm that satisfy four requirements: 1. The measurement ensemble succeeds for all signals, with high probability over the random choices in its construction. 2. The number of measurements of the signal is optimal, except for a factor polylogarithmic in the signal length. 3. The running time of the algorithm is polynomial in the amount of information in the signal and polylogarithmic in the signal length. 4. The recovery algorithm offers the strongest possible type of error guarantee. Moreover, it is a fully polynomial approximation scheme with respect to this type of error bound. Emerging applications demand this level of performance. Yet no other algorithm in the literature simultaneously achieves all four of these desiderata.

### Citations

1715 | Compressed sensing - Donoho - 2006 |

831 | Practical Signal Recovery from Random
- Candes, Romberg, et al.
(Show Context)
Citation Context ... For clarity, we focus on the case ε = 1. To obtain an approximation scheme, we substitute m/ε 2 for m, which increases the costs by (1/ε) O(1) .Approach, Refs Error bd. # Meas. Time ℓ1 min. + Gauss =-=[3]-=- ℓ1 min. + Fourier Combinatorial [5] Combinatorial [5] Chaining Pursuit [10] HHS (this result) ‚ ‚E ‚ ≤ m 2 −1/2‚ ‚Eopt‚ 1 m log(d/m) LP(md) ‚ ‚ E ‚2 ≤ m−1/2‚ ‚ Eopt‚ 1 m log4 d d log d (empirical) ‚ ... |

741 |
Stable signal recovery from incomplete and inaccurate measurements
- Candès, Romberg, et al.
- 2006
(Show Context)
Citation Context ...rem. Corollary 2. Let b fm be the best m-term approximation to the output b f of HHS Pursuit. Then ‚ ‚ f − fm b ‚ 2 ≤ ‖f − fm‖ 2 + 2ε √ m ‖f − fm‖ 1 . This result should be compared with Theorem 2 of =-=[2]-=-, which gives an analogous bound for the (superlinear) ℓ1 minimization algorithm. A second corollary provides an ℓ1 error estimate. Corollary 3. Let b fm be the best m-term approximation to the output... |

295 | Signal recovery from random measurements via orthogonal matching pursuit
- Tropp, Gilbert
- 2007
(Show Context)
Citation Context ...probability, on all signals, the algorithm succeeds.” Most approaches to Compressed Sensing yield uniform guarantees— exceptions include work on Orthogonal Matching Pursuit (OMP) due to Tropp–Gilbert =-=[16]-=- and the randomized algorithm of Cormode–Muthukrishnan [5] which achieves the strongest error bounds. Our algorithm achieves a uniform bound because, unlike “for each” algorithms, HHS uses a stronger ... |

151 | Compressed sensing and best k-term approximation
- Cohen, Wolfgang, et al.
- 2009
(Show Context)
Citation Context ...≤ (1 + ε) ‖f − fm‖ 2 . It has been established [9] that this guarantee is possible if we construct a random measurement matrix for each signal. On the other hand, Cohen, Dahmen, and DeVore have shown =-=[4]-=- that it is impossible to obtain this error bound simultaneously for all signals unless the number of measurements is Ω(d). The same authors also proved a more general lower bound [4]. For each p in t... |

138 | Data compression and harmonic analysis
- Donoho, Vetterli, et al.
- 1998
(Show Context)
Citation Context ...erty that its components decay when sorted by magnitude. These signals arise in numerous applications because one can compress the wavelet and Fourier expansions of certain classes of natural signals =-=[7]-=-. A common measure of compressibility is the weak-ℓp norm, which is defined for 0 < p < ∞ as ‖f‖ wℓp def = inf{r : |f|(k) ≤ r · k −1/p for k = 1, 2, . . . , d}.The notation |f| (k) indicates the kth ... |

95 |
On the fast Fourier transform of functions with singularities
- Beylkin
- 1995
(Show Context)
Citation Context ... large component, and the remaining components have ℓ1 norm smaller than the magnitude of the large component. of any nontrivial algorithm for this problem, despite the existence of faster algorithms =-=[1]-=- for problems that are superficially similar. We encode the recovered spikes by accessing the columns of the identification and estimation matrices corresponding to the locations of these spikes and t... |

79 | Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements, preprint
- Rudelson, Vershynin
(Show Context)
Citation Context ... close to the residual, even though it contains O(M √ log M) nonzero entries. There are two technical results that are essential to the proof. The first appeared in the work of Rudelson and Vershynin =-=[14]-=-. Lemma 14 (Restricted Isometry). The estimation matrix Φ has the property that every |L|-column submatrix A satisfies for every vector x, 1 2 ‖x‖ 2 ≤ ‖Ax‖ 2 ≤ 3 2 ‖x‖ 2 . Lemma 15. The estimation mat... |

69 | A new compressive imaging camera architecture using optical-domain compression
- Takhar, Laska, et al.
- 2006
(Show Context)
Citation Context ...the time it takes to reconstruct signals from measurements. Scientists and engineers are developing technologies where the computational model of Compressed Sensing applies. They are building cameras =-=[15, 18]-=-, analog-to-digital converters [8, 13, 12], and other sensing devices [20, 19] that can obtain a general linear measurement of a signal at unit cost. Compressive imaging cameras use a digital micro-mi... |

64 | Combinatorial algorithms for compressed sensing
- Cormode, Muthukrishnan
- 2006
(Show Context)
Citation Context ...pt = Eopt(f) = f − fm denote the error vector ‚ for the optimal output. Also, let Copt,p denote maxg ‚Eopt(g) ‚ , p where g is the worst possible signal in the class where f lives. The two results in =-=[5]-=- refer to the two deterministic constructions which are uniform on a class of functions (noted in the result). LP(md) denotes resources needed to solve a linear program with Θ(md) variables, plus mino... |

48 | Improved time bounds for near-optimal sparse fourier representation via sampling
- Gilbert, Muthukrishnan, et al.
- 2005
(Show Context)
Citation Context ...might hope to take (m/ε 2 ) polylog(d) measurements of a signal f and produce an m-sparse approximation bf that satisfies the error bound ‚ f − b f ‚ ‚2 ≤ (1 + ε) ‖f − fm‖ 2 . It has been established =-=[9]-=- that this guarantee is possible if we construct a random measurement matrix for each signal. On the other hand, Cohen, Dahmen, and DeVore have shown [4] that it is impossible to obtain this error bou... |

43 | Random sampling for analog-to-information conversion of wideband signals
- Laska, Kirolos, et al.
- 2006
(Show Context)
Citation Context ...s from measurements. Scientists and engineers are developing technologies where the computational model of Compressed Sensing applies. They are building cameras [15, 18], analog-to-digital converters =-=[8, 13, 12]-=-, and other sensing devices [20, 19] that can obtain a general linear measurement of a signal at unit cost. Compressive imaging cameras use a digital micro-mirror ar-ray to optically compute inner pr... |

43 | Compressive imaging for video representation and coding
- Wakin, Laska, et al.
- 2006
(Show Context)
Citation Context ...the time it takes to reconstruct signals from measurements. Scientists and engineers are developing technologies where the computational model of Compressed Sensing applies. They are building cameras =-=[15, 18]-=-, analog-to-digital converters [8, 13, 12], and other sensing devices [20, 19] that can obtain a general linear measurement of a signal at unit cost. Compressive imaging cameras use a digital micro-mi... |

41 |
Fast reconstruction of piecewise smooth signals from random projections
- Duarte, Wakin, et al.
- 2005
(Show Context)
Citation Context ...s from measurements. Scientists and engineers are developing technologies where the computational model of Compressed Sensing applies. They are building cameras [15, 18], analog-to-digital converters =-=[8, 13, 12]-=-, and other sensing devices [20, 19] that can obtain a general linear measurement of a signal at unit cost. Compressive imaging cameras use a digital micro-mirror ar-ray to optically compute inner pr... |

37 | Analog-to-information conversion via random demodulation
- Kirolos, Laska, et al.
- 2006
(Show Context)
Citation Context ...s from measurements. Scientists and engineers are developing technologies where the computational model of Compressed Sensing applies. They are building cameras [15, 18], analog-to-digital converters =-=[8, 13, 12]-=-, and other sensing devices [20, 19] that can obtain a general linear measurement of a signal at unit cost. Compressive imaging cameras use a digital micro-mirror ar-ray to optically compute inner pr... |

25 | Algorithmic linear dimension reduction in the ℓ1 norm for sparse vectors
- Gilbert, Strauss, et al.
- 2006
(Show Context)
Citation Context ...ee below) tailored to the mixednorm bound of Theorem 1. (We include in Table 1 uniform results only.) Chaining Pursuit is the only algorithm in the literature that achieves the first three desiderata =-=[10]-=-. The error bound in Chaining Pursuit, however, is less than optimal. Not only is this error bound worse than the HHS error bound, but also Chaining Pursuit is not an approximation scheme. Our algoirt... |

14 | Large deviation inequalities for sums of indicator variables
- Janson
- 1994
(Show Context)
Citation Context ...o understand large deviations of X requires some effort because the set of indicators {Xn} is not stochastically independent. Nevertheless, X satisfies a rather strong tail bound. Fact 11 (Theorem 6, =-=[11]-=-). n P {X ≥ E X + a} ≤ exp − ` σ 2 + a ´ “ log 1 + a σ2 ” o − a . This result is based on the surprising fact, due to Vatutin and Mikhailov [17], that X can be expressed as a sum of independent indica... |

7 |
Fiber-optic localization by geometric space coding with a two-dimensional gray code
- Zheng, Brady, et al.
- 2005
(Show Context)
Citation Context ...ineers are developing technologies where the computational model of Compressed Sensing applies. They are building cameras [15, 18], analog-to-digital converters [8, 13, 12], and other sensing devices =-=[20, 19]-=- that can obtain a general linear measurement of a signal at unit cost. Compressive imaging cameras use a digital micro-mirror ar-ray to optically compute inner products of the image with pseudorando... |

4 | Nonadaptive group testing based fiber sensor deployment for multiperson tracking
- Zheng, Pitsianis, et al.
(Show Context)
Citation Context ...ineers are developing technologies where the computational model of Compressed Sensing applies. They are building cameras [15, 18], analog-to-digital converters [8, 13, 12], and other sensing devices =-=[20, 19]-=- that can obtain a general linear measurement of a signal at unit cost. Compressive imaging cameras use a digital micro-mirror ar-ray to optically compute inner products of the image with pseudorando... |

3 |
Limit theorems for the numer of empty cells in an equiprobable scheme for group allocation of particles. Theor
- Vatutin, Mikhailov
- 1982
(Show Context)
Citation Context ...tisfies a rather strong tail bound. Fact 11 (Theorem 6, [11]). n P {X ≥ E X + a} ≤ exp − ` σ 2 + a ´ “ log 1 + a σ2 ” o − a . This result is based on the surprising fact, due to Vatutin and Mikhailov =-=[17]-=-, that X can be expressed as a sum of independent indicators. The content of our argument is to develop explicit bounds on the expectation and variance of X, which will allow us to apply Janson’s resu... |