Sparsity in Time-Frequency Representations

Pfander, Götz E.; Rauhut, Holger

doi:10.1007/s00041-009-9086-9

Cited by 65 publications

(39 citation statements)

References 36 publications

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“…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). A non-uniform recovery result for Gabor synthesis matrices with Steinhaus generator (see Section 2 for the definition) appears in [32], where it was shown that a fixed s-sparse vector is recovered from its image under a random draw of the m × m 2 Gabor synthesis matrix via ℓ 1 -minimization with high probability provided that m ≥ Cs log m. Again, the conclusion of Theorem 1.3 is stronger than this previous result in the sense that it implies uniform and stable s-sparse recovery. Further related material may be found in [33,2].…”

Section: Theorem 13 Let ǫ Be a Rademacher Vector And Consider The Gamentioning

confidence: 99%

Suprema of Chaos Processes and the Restricted Isometry Property

Krahmer

Mendelson

Rauhut

2014

Comm Pure Appl Math

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280

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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: Theorem 13 Let ǫ Be a Rademacher Vector And Consider The Gamentioning

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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“…The matrix Ψ g ∈ C n×n 2 whose columns list the members π(λ)g, λ ∈ Z n ×Z n , of the Gabor system is referred to as Gabor synthesis matrix [16,32,40]. Note that Ψ g allows for fast matrix vector multiplication algorithms based on the FFT.…”

Section: Time-frequency Structured Measurement Matricesmentioning

confidence: 99%

The restricted isometry property for time–frequency structured random matrices

Pfander

Rauhut

Tropp

2012

Probab. Theory Relat. Fields

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We establish the restricted isometry property for finite dimensional Gabor systems, that is, for families of time-frequency shifts of a randomly chosen window function. We show that the s-th order restricted isometry constant of the associated n × n 2 Gabor synthesis matrix is small provided s ≤ c n 2/3 / log 2 n. This improves on previous estimates that exhibit quadratic scaling of n in s. Our proof develops bounds for a corresponding chaos process.

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“…A large body of literature extends these results to measurements of signals in the presence of noise, to signals that are not exactly sparse but compressible, 5 to several types of measurement matrices 6,[21][22][23]27 and to measurement models beyond simple sparsity. 2 …”

Section: Compressed Sensing Backgroundmentioning

confidence: 99%

Compressed sensing for fusion frames

2009

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Compressed Sensing (CS) is a new signal acquisition technique that allows sampling of sparse signal using significantly fewer measurements than previously thought possible. On the other hand, a fusion frame is a new signal representation method that uses collections of subspaces instead of vectors to represent signals. This work combines these exciting new fields to introduce a new sparsity model for fusion frames. Signals that are sparse under the new model can be compressively sampled and uniquely reconstructed in ways similar to sparse signals using standard CS. The combination provides a promising new set of mathematical tools and signal models useful in a variety of applications.With the new model, a sparse signal has energy in very few of the subspaces of the fusion frame, although it needs not be sparse within each of the subspaces it occupies. We define a mixed 1 / 2 norm for fusion frames. A signal sparse in the subspaces of the fusion frame can thus be sampled using very few random projections and exactly reconstructed using a convex optimization that minimizes this mixed 1 / 2 norm. The sampling conditions we derive are very similar to the coherence and RIP conditions used in standard CS theory.

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Sparsity in Time-Frequency Representations

Cited by 65 publications

References 36 publications

Suprema of Chaos Processes and the Restricted Isometry Property

Suprema of Chaos Processes and the Restricted Isometry Property

The restricted isometry property for time–frequency structured random matrices

Compressed sensing for fusion frames

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