The restricted isometry property for time–frequency structured random matrices

Pfander, Götz E.; Rauhut, Holger; Tropp, Joel A.

doi:10.1007/s00440-012-0441-4

Cited by 46 publications

(32 citation statements)

References 36 publications

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“…The following formulation again focuses on normalized Rademacher vectors, postponing a more general version of our results until Section 5. Again, Theorem 1.3 improves the best previously known estimate from [35], in which the sufficient condition of m C s 3=2 log 3 m was derived. In particular, it implies the first uniform sparse recovery result with a linear scaling of the number of samples m in the sparsity s (up to log-factors).…”

Section: Time-frequency Structured Random Matricesmentioning

confidence: 99%

Suprema of Chaos Processes and the Restricted Isometry Property

Krahmer

Mendelson

Rauhut

2014

Comm Pure Appl Math

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221

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We present a new bound for suprema of a special type of chaos processes indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and time-frequency structured random matrices. In both cases the required condition on the number m of rows in terms of the sparsity s and the vector length n is m s log 2 s log 2 n.

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Section: Time-frequency Structured Random Matricesmentioning

confidence: 99%

Suprema of Chaos Processes and the Restricted Isometry Property

Krahmer

Mendelson

Rauhut

2014

Comm Pure Appl Math

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221

280

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“…typically a real-valued, smooth function of good decay, symmetric around the origin (hence with a Fourier transform with similar properties), but the definition of the DGT imposes no restrictions on g. This property has been used to lower the computational complexity of sparse compression techniques [28,29]. The DGT is equivalent to a Fourier modulated filter bank with M channels and decimation in time a [7], and it is a valuable tool for time-frequency analysis when a linear frequency scale is desired.…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for the Discrete Gabor Transform with a Long Fir Window

Søndergaard

2011

J Fourier Anal Appl

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The Discrete Gabor Transform (DGT) is the most commonly used signal transform for signal analysis and synthesis using a linear frequency scale. The development of the Linear Time-Frequency Analysis Toolbox (LTFAT) has been based on a detailed study of many variants of the relevant algorithms. As a side result of these systematic developments of the subject, two new methods are presented here. Comparisons are made with respect to the computational complexity, and the running time of optimised implementations in the C programming language. The new algorithms have the lowest known computational complexity and running time when a long FIR window is used. The implementations are freely available for download. By summarizing general background information on the state of the art, this article can also be seen as a research survey, sharing with the readers experience in the numerical work in Gabor analysis.

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“…, n − 1} and the floor and ceiling operation for x ∈ R is denoted by ⌊x⌋ respectively ⌈x⌉. By Σ n s we denote all s−sparse vectors x in R n respectively C n , i.e., satisfying supp(x) ≤ s. For recovery results for unknown x ∈ Σ n s with y not sparse and randomly chosen, see [19,10]; for the related time-varying setting see [15,16,17,10].…”

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confidence: 99%

On the stability of sparse convolutions

Walk

Jung

Pfander

2017

Applied and Computational Harmonic Analysis

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We give a stability result for sparse convolutions on 2 (G) × 1 (G) for torsion-free discrete Abelian groups G such as Z. It turns out, that the torsion-free property prevents full cancellation in the convolution of sparse sequences and hence allows to establish stability in each entry, that is, for any fixed entry of the convolution the resulting linear map is injective with an universal lower norm bound, which only depends on the support cardinalities of the sequences. This can be seen as a reverse statement of the famous Young inequality for sparse convolutions. Our result hinges on a compression argument in additive set theory. I. INTRODUCTIONAdditive problems have increasingly become a focus in combinatorics, number theory, group theory and Fourier analysis as pointed out, e.g., in the textbook of Tao and Vu [24]. The key hereby is an understanding of the additive structure of finite subsets of an Abelian group G. The main result in the herein presented work is the application of a recent compression result in additive set theory by Grynkiewicz [6, Theorem 20.10] to sparse convolutions on discrete Abelian groups which are torsionfree, i.e., for any N ∈ N and g ∈ G it holds N g = g + ⋅ ⋅ ⋅ + g = 0 if and only if g = 0. Compressing the convolution of sparse sequences, i.e., sequences with finite support sets, reduces to a compression of the sumset of their supports, since supp(x * y) ⊆ supp(x)+supp(y). Our compression result allows to obtain a reverse statement of Young's inequality [28,27] for the convolution of all sparse (x, y) ∈ 2 (G)×

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The restricted isometry property for time–frequency structured random matrices

Cited by 46 publications

References 36 publications

Suprema of Chaos Processes and the Restricted Isometry Property

Suprema of Chaos Processes and the Restricted Isometry Property

Efficient Algorithms for the Discrete Gabor Transform with a Long Fir Window

On the stability of sparse convolutions

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