## Deterministic constructions of compressed sensing matrices (2007)

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Venue: | List of References November 2001 B.1 Complete IDL Listing OMG IDL B // File: CosActivity |

Citations: | 88 - 2 self |

### BibTeX

@TECHREPORT{Devore07deterministicconstructions,

author = {Ronald A. Devore},

title = {Deterministic constructions of compressed sensing matrices},

institution = {List of References November 2001 B.1 Complete IDL Listing OMG IDL B // File: CosActivity},

year = {2007}

}

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### Abstract

With high esteem to Professor Henryk Wozniakowski on the occasion of his 60-th birthday Compressed sensing is a new area of signal processing. Its goal is to minimize the number of samples that need to be taken from a signal for faithful reconstruction. The performance of compressed sensing on signal classes is directly related to Gelfand widths. Similar to the deeper constructions of optimal subspaces in Gelfand widths, most sampling algorithms are based on randomization. However, for possible circuit implementation, it is important to understand what can be done with purely deterministic sampling. In this note we show how to construct sampling matrices using finite fields. One such construction gives cyclic matrices which are interesting for circuit implementation. While the guaranteed performance of these deterministic constructions is not comparable to the random constructions, these matrices have the best known performance for purely deterministic constructions. 1

### Citations

1710 | Compressed sensing
- Donoho
- 2006
(Show Context)
Citation Context ...CoR N00014-05-1-0715; and the National Science Foundation Grant DMS-354707. 1smethodology of Compressed Sensing traces back to early work on Gelfand widths and Information Based Complexity (IBC); see =-=[6, 5, 3]-=- for a discussion of these connections. This paper will be concerned with the discrete compressed sensing problem where we are given a discrete signal which is a vector x ∈ IR N with N large and we wi... |

652 | Decoding by linear programming
- Candes, Tao
- 2005
(Show Context)
Citation Context ...is that they can be more readily implemented in circuits. An outline of our paper is the following. In the next section, we discuss the Restricted Isometry Property (RIP) introduced by Candes and Tao =-=[4]-=- and how this property guarantees upper bounds for the performance of CS matrices on classes. The following section, gives our construction of CS matrices and the proof that they satisfy a RIP. The fi... |

151 | Compressed sensing and best k-term approximation
- Cohen, Wolfgang, et al.
- 2009
(Show Context)
Citation Context ...CoR N00014-05-1-0715; and the National Science Foundation Grant DMS-354707. 1smethodology of Compressed Sensing traces back to early work on Gelfand widths and Information Based Complexity (IBC); see =-=[6, 5, 3]-=- for a discussion of these connections. This paper will be concerned with the discrete compressed sensing problem where we are given a discrete signal which is a vector x ∈ IR N with N large and we wi... |

25 |
The widths of certain finite dimensional sets and classes of smooth functions
- Kashin
- 1977
(Show Context)
Citation Context ... result in this field. It states that there exist absolute constants C1, C2 such that � log(N/n) C1 ≤ d n n (U(ℓ N 1 )) ℓN � log(N/n) ≤ C2 . (1.8) 2 n The upper estimate in (1.8) was proved by Kashin =-=[8]-=- save for the correct power of the logarithm. Later Garneev and Gluskin proved the upper and lower bounds in (1.8) (see [7]). The upper bound is proved via random constructions and there remains to th... |

14 |
Norms of random matrices and widths of finite-dimensional sets
- Gluskin
- 1984
(Show Context)
Citation Context ...� log(N/n) ≤ C2 . (1.8) 2 n The upper estimate in (1.8) was proved by Kashin [8] save for the correct power of the logarithm. Later Garneev and Gluskin proved the upper and lower bounds in (1.8) (see =-=[7]-=-). The upper bound is proved via random constructions and there remains to this date no deterministic proof of the upper bound in (1.8). In compressed sensing, their constructions correspond to random... |

3 |
K.: Mixed norm n-widths
- Boor, DeVore, et al.
- 1980
(Show Context)
Citation Context ...width of classes with such constructions. We shall give constructions of matrices Φ using finite fields which are related to the use of finite fields to prove results on Kolmogorov widths as given in =-=[2]-=-. A related construction using number theory was given by Maiorov [9]. Our constructions 3swill not give optimal or near optimal performance, as will be explained later. However their performance is t... |

2 |
The Johnson–Lindenstrauss meets compressed sensing
- Baraniuk, Davenport, et al.
(Show Context)
Citation Context .... (2.5) To get the optimal result we want Φ to satisfy RIP of order k = n/ log(N/n). Matrices of this type can be constructed using random variables such as Gaussian or Bernouli as their entries (see =-=[1]-=- for example). However, there are no deterministic constructions for k of this size. In the next section, we shall give a deterministic construction of matrices Φ which satisfy RIP for a more modest r... |

2 |
Optimal Computation, vol
- DeVore
(Show Context)
Citation Context ...CoR N00014-05-1-0715; and the National Science Foundation Grant DMS-354707. 1smethodology of Compressed Sensing traces back to early work on Gelfand widths and Information Based Complexity (IBC); see =-=[6, 5, 3]-=- for a discussion of these connections. This paper will be concerned with the discrete compressed sensing problem where we are given a discrete signal which is a vector x ∈ IR N with N large and we wi... |

2 |
Linear diameters of Sobolev classes, Soviet Dokl
- Maiorov
- 1991
(Show Context)
Citation Context ... of matrices Φ using finite fields which are related to the use of finite fields to prove results on Kolmogorov widths as given in [2]. A related construction using number theory was given by Maiorov =-=[9]-=-. Our constructions 3swill not give optimal or near optimal performance, as will be explained later. However their performance is the best known to the author for deterministic constructions. We shall... |

2 |
Trigonometric widths of Sobolev classes in the space
- Maiorov
- 1986
(Show Context)
Citation Context ... using finite fields which are related to the use of finite fields to prove results on Kolmogorov widths as given in [2]. A related construction using number theory was given by Maiorov [9] (see also =-=[10]-=- 3sfor another deterministic construction). Our constructions will not give optimal or near optimal performance, as will be explained later. However their performance is the best known to the author f... |

1 |
Constructive Approximation:Advanced Problems
- Makovoz
- 1996
(Show Context)
Citation Context ...e have gotten on k appear to be the best we could expect to get by this approach. Indeed, with an eye toward results on distribution of scalar products of unit vectors (see Lemma 4.1 of Chapter 14 in =-=[10]-=-), it seems that we could not improve much on the bounds we gave for diagonal dominance. Of course, the spectral norm of a matrix can be much smaller than the ℓ1, ℓ∞ norms. Thus it may be that estimat... |

1 |
Linear diameters of Sobolev classes
- Maiorov
(Show Context)
Citation Context ... of matrices Φ using finite fields which are related to the use of finite fields to prove results on Kolmogorov widths as given in [2]. A related construction using number theory was given by Maiorov =-=[9]-=- (see also [10] 3sfor another deterministic construction). Our constructions will not give optimal or near optimal performance, as will be explained later. However their performance is the best known ... |