## Explicit constructions for compressed sensing of sparse signals (2008)

Venue: | In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms |

Citations: | 38 - 3 self |

### BibTeX

@INPROCEEDINGS{Indyk08explicitconstructions,

author = {Piotr Indyk},

title = {Explicit constructions for compressed sensing of sparse signals},

booktitle = {In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms},

year = {2008},

pages = {30--33}

}

### OpenURL

### Abstract

Over the recent years, a new approach for obtaining a succinct approximate representation of ndimensional

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Citation Context ... of a signal x by using only (k + log n) O(1) dot products of (inverse) Fourier matrix rows with the signal. Amazingly enough, this implication has not been explicitly stated until the recent work of =-=[CT06]-=-. 3 The constant E depends on the best-known construction of an extractor. 2sdistributes the non-zero elements of the vector x in such a way that “most” elements are mapped to non-overflowing buckets ... |

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Citation Context ...y the notion of row tensor product [CM06, GSTV06] and the explicit construction of [CM06]. 1sHow much information about signal x can be retrieved from its sketch Ax ? As it turns out, quite a lot. In =-=[AMS96]-=- (cf. [JL84]) it was shown that one can retrieve the approximation to �x�2 from a very short sketch Ax. Further developments [GGI + 02a] resulted in algorithms which, given a sketch of length R = (k +... |

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Citation Context ...of row tensor product [CM06, GSTV06] and the explicit construction of [CM06]. 1sHow much information about signal x can be retrieved from its sketch Ax ? As it turns out, quite a lot. In [AMS96] (cf. =-=[JL84]-=-) it was shown that one can retrieve the approximation to �x�2 from a very short sketch Ax. Further developments [GGI + 02a] resulted in algorithms which, given a sketch of length R = (k + log n) O(1)... |

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Citation Context ...n of this document has been greatly influenced (and simplified) thanks to the recent developments in the area, notably the notion of row tensor product [CM06, GSTV06] and the explicit construction of =-=[CM06]-=-. 1sHow much information about signal x can be retrieved from its sketch Ax ? As it turns out, quite a lot. In [AMS96] (cf. [JL84]) it was shown that one can retrieve the approximation to �x�2 from a ... |

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Citation Context ...d basis e1 . . . en; the sketching matrix A can be adapted to any other basis by multiplying it by the change-of-basis matrix. 2 The origin of these results can be traced even further in the past, to =-=[Man92]-=- (cf. [GGI + 02b, GMS05]). In [Man92, GGI + 02a], it was shown how to estimate the largest k Fourier coefficients of a given signal vector by accessing only (k + log n) O(1) entries of the vector. By ... |

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Citation Context ...ributes the non-zero elements of the vector x in such a way that “most” elements are mapped to non-overflowing buckets (this part of the construction resembles the use of extractors in a recent paper =-=[Ind07]-=- by the author). The elements in non-overflowing buckets are then recovered using group testing techniques. Unfortunately, the recovery process is not complete, due to the elements in the overflowing ... |

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Citation Context ...ems: (i) either they require sketches of size at least quadratic in r [CM06, Mut06, DeV07], or (ii) representing matrix entries requires Ω(n) bits per entry (when van der Monde matrices are used, see =-=[DeV06]-=-, p. 20). See [Tao07] for a discussion on explicit construction for compressed sensing. Result. In this paper we provide an explicit construction of a binary R×n matrix A, such that one can recover an... |

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Citation Context ...ng a well-investigated method of Basis Pursuit. Faster (in fact, sublinear-time) algorithms for uniform recovery were discovered in [CM06, GSTV06, GSTV07]. Many other papers are devoted to this topic =-=[Gro06]-=-. Several of the aforementioned algorithms address a particularly simple class of pure signals, where the signal x is r-sparse, that is, it contains only r non-zero entries. In this case the aforement... |

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Citation Context ...f all edges from v to X. 3 Proof We start by selecting a bipartite graph G = (A, B, E), where A = [n] and B = [k]. The nodes in A have degree dA. The graph is chosen so that it is an ǫ-extractor (see =-=[Sha04]-=- for an overview) for some ǫ > 0 specified later. That is, G has the following property. For any X ⊂ A, |X| ≥ k, consider the distribution D over B obtained by choosing a ∈ X uniformly at random, and ... |

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1 |
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Citation Context ...or any absolute constant ǫ > 0, ǫ-extractors can be explicitly constructed, with dA = 2O(log log n)E, for a constant E > 1. It appears that the best known extractor construction guarantees E = 2 (see =-=[Vad07]-=-, Lecture 12, Table 1). For an integer t > 0, define Overflowt(X) = {b ∈ B : |Γ(b, X)| > t} In the following we set t = 2dA. Observe that, for any b ∈ Overflowt(X), we have PrD[b] ≥ 2/|B|. It follows ... |