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Dive into the research topics where Holger Rauhut is active.

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Featured researches published by Holger Rauhut.


IEEE Transactions on Information Theory | 2008

Compressed Sensing and Redundant Dictionaries

Holger Rauhut; Karin Schnass; Pierre Vandergheynst

This paper extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry constants. Thus, signals that are sparse with respect to the dictionary can be recovered via basis pursuit (BP) from a small number of random measurements. Further, thresholding is investigated as recovery algorithm for compressed sensing, and conditions are provided that guarantee reconstruction with high probability. The different schemes are compared by numerical experiments.


IEEE Transactions on Information Theory | 2010

Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation

Yonina C. Eldar; Holger Rauhut

This paper considers recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worst case analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. Here, the main focus is on analysis of convex relaxation techniques. In particular, the mixed l 2,1 approach to multichannel recovery is investigated. Under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, it is shown that the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. The probability bounds are valid and meaningful even for a small number of signals. Using the tools developed to analyze the convex relaxation technique, also previous bounds for thresholding and SOMP recovery are tightened.


SIAM Journal on Numerical Analysis | 2008

Recovery Algorithms for Vector-Valued Data with Joint Sparsity Constraints

Massimo Fornasier; Holger Rauhut

Vector-valued data appearing in concrete applications often possess sparse expansions with respect to a preassigned frame for each vector component individually. Additionally, different components may also exhibit common sparsity patterns. Recently, there were introduced sparsity measures that take into account such joint sparsity patterns, promoting coupling of nonvanishing components. These measures are typically constructed as weighted


IEEE Journal of Selected Topics in Signal Processing | 2010

Compressive Estimation of Doubly Selective Channels in Multicarrier Systems: Leakage Effects and Sparsity-Enhancing Processing

Georg Tauböck; Franz Hlawatsch; Daniel Eiwen; Holger Rauhut

\ell_1


Foundations of Computational Mathematics | 2008

Random Sampling of Sparse Trigonometric Polynomials, II. Orthogonal Matching Pursuit versus Basis Pursuit

Stefan Kunis; Holger Rauhut

norms of componentwise


IEEE Transactions on Information Theory | 2008

Stability Results for Random Sampling of Sparse Trigonometric Polynomials

Holger Rauhut

\ell_q


Siam Journal on Optimization | 2011

Low-rank Matrix Recovery via Iteratively Reweighted Least Squares Minimization

Massimo Fornasier; Holger Rauhut; Rachel Ward

norms of frame coefficients. We show how to compute solutions of linear inverse problems with such joint sparsity regularization constraints by fast thresholded Landweber algorithms. Next we discuss the adaptive choice of suitable weights appearing in the definition of sparsity measures. The weights are interpreted as indicators of the sparsity pattern and are iteratively updated after each new application of the thresholded Landweber algorithm. The resulting two-step algorithm is interpreted as a double-minimization scheme for a suitable target functional. We show its


Journal of Complexity | 2010

The Gelfand widths of l p -balls for 0<p≤1

Simon Foucart; Alain Pajor; Holger Rauhut; Tino Ullrich

\ell_2


IEEE Transactions on Information Theory | 2011

Sparse Recovery From Combined Fusion Frame Measurements

Petros T. Boufounos; Gitta Kutyniok; Holger Rauhut

-norm convergence. An implementable version of the algorithm is also formulated, and its norm convergence is proven. Numerical experiments in color image restoration are presented.


IEEE Transactions on Signal Processing | 2008

Identification of Matrices Having a Sparse Representation

Goetz E. Pfander; Holger Rauhut; Jared Tanner

We consider the application of compressed sensing (CS) to the estimation of doubly selective channels within pulse-shaping multicarrier systems (which include orthogonal frequency-division multiplexing (OFDM) systems as a special case). By exploiting sparsity in the delay-Doppler domain, CS-based channel estimation allows for an increase in spectral efficiency through a reduction of the number of pilot symbols. For combating leakage effects that limit the delay-Doppler sparsity, we propose a sparsity-enhancing basis expansion and a method for optimizing the basis with or without prior statistical information about the channel. We also present an alternative CS-based channel estimator for (potentially) strongly time-frequency dispersive channels, which is capable of estimating the ¿off-diagonal¿ channel coefficients characterizing intersymbol and intercarrier interference (ISI/ICI). For this estimator, we propose a basis construction combining Fourier (exponential) and prolate spheroidal sequences. Simulation results assess the performance gains achieved by the proposed sparsity-enhancing processing techniques and by explicit estimation of ISI/ICI channel coefficients.

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Gitta Kutyniok

Technical University of Berlin

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Rachel Ward

University of Texas at Austin

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Karin Schnass

École Polytechnique Fédérale de Lausanne

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Pierre Vandergheynst

École Polytechnique Fédérale de Lausanne

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