## Just relax: Convex programming methods for subset selection and sparse approximation (2004)

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@TECHREPORT{Tropp04justrelax:,

author = {Joel A. Tropp},

title = {Just relax: Convex programming methods for subset selection and sparse approximation},

institution = {},

year = {2004}

}

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### Abstract

Abstract. Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical engineering, applied mathematics and statistics, but small theoretical progress has been made over the last fifty years. Subset selection and sparse approximation both admit natural convex relaxations, but the literature contains few results on the behavior of these relaxations for general input signals. This report demonstrates that the solution of the convex program frequently coincides with the solution of the original approximation problem. The proofs depend essentially on geometric properties of the ensemble of elementary signals. The results are powerful because sparse approximation problems are combinatorial, while convex programs can be solved in polynomial time with standard software. Comparable new results for a greedy algorithm, Orthogonal Matching Pursuit, are also stated. This report should have a major practical impact because the theory applies immediately to many real-world signal processing problems. 1.

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Citation Context ...ter in terms of the dimension d and the number of atoms N [SH03]: � 0 N ≤ d µ ≥ N > d. � N−d d (N−1) If N > d and the dictionary contains an orthonormal basis, the lower bound increases to µ ≥ 1/ √ d =-=[GN03b]-=-. Although the coherence exhibits a quantum jump as soon as the number of atoms exceeds the dimension, it is possible to construct very large dictionaries with low coherence. When d = 2 k , Calderbank... |

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Citation Context ...ave unit norm, dist(z, w) = � 1 − |〈z, w〉| 2 . Evidently, the distance between two lines ranges between zero and one. Equipped with this metric, P d−1 (C) forms a smooth, compact, Riemannian manifold =-=[CHS96]-=-. 3.9. Minimum Distance, Maximum Correlation. We view the dictionary D as a finite set of lines in the projective space P d−1 (C). Given an arbitary nonzero signal s, we shall calculate the minimum di... |

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Citation Context ...putations [CDS99]. Sardy, Bruce, and Tseng have also written a paper on ands36 J. A. TROPP using cyclic minimization to solve the Basis Pursuit problem when the dictionary has a block-basis structure =-=[SBT00]-=-. Starck, Donoho, and Candès have proposed an iterative method for solving (A.1) when the dictionary has block-basis structure [SDC03]. Most recently, Daubechies, Defrise, and De Mol have developed an... |

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Citation Context ...utions to both problems. ⋆ ⋆ ⋆ ⋆ ⋆ Sparse approximation has been studied for nearly a century, and it has numerous applications. Temlyakov [Tem02] locates the first example in a 1907 paper of Schmidt =-=[Sch07]-=-. In the 1950s, statisticians launched an extensive investigation of another sparse approximation problem called subset selection in regression [Mil02]. Later, approximation theorists began a systemat... |

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Citation Context ...struct very large dictionaries with low coherence. When d = 2 k , Calderbank et al. have produced a striking example of a dictionary that contains (d+1) orthonormal bases yet retains coherence 1/ √ d =-=[CCKS97]-=-. Gilbert, Muthukrishnan and Strauss have exhibited a method for constructing even larger dictionaries with slightly higher coherence [GMS03].s10 J. A. TROPP The coherence parameter of a dictionary wa... |

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Citation Context ...ression would measure diversity as the number of bits necessary to represent the coefficient vector with a certain precision. Gribonval and Nielsen have made some preliminary progress on this problem =-=[GN03a]-=-. Second, one may wish to consider sparse approximation problems with respect to error measures other than the usual Euclidean distance. For example, the uniform norm would promote sparse approximatio... |

54 |
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