Holger Rauhut
RWTH Aachen University
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Publication
Featured researches published by Holger Rauhut.
IEEE Transactions on Information Theory | 2008
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
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
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
Georg Tauböck; Franz Hlawatsch; Daniel Eiwen; Holger Rauhut
\ell_1
Foundations of Computational Mathematics | 2008
Stefan Kunis; Holger Rauhut
norms of componentwise
IEEE Transactions on Information Theory | 2008
Holger Rauhut
\ell_q
Siam Journal on Optimization | 2011
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
Simon Foucart; Alain Pajor; Holger Rauhut; Tino Ullrich
\ell_2
IEEE Transactions on Information Theory | 2011
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
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.