arXiv subjects

Holger Rauhut

Publications and source records attributed to Holger Rauhut.

At least 73 records · Page 4Linked to original sources

arXiv2013

Fast and RIP-optimal transforms

We study constructions of $k \times n$ matrices $A$ that both (1) satisfy the restricted isometry property (RIP) at sparsity $s$ with optimal parameters, and (2) are efficient in the sense that only $O(n\log n)$ operations are required to compute $Ax$ given a vector $x$. Our construction is based on repeated application of independent transformations of the form $DH$, where $H$ is a Hadamard or Fourier transform and $D$ is a diagonal matrix with random $\{+1,-1\}$ elements on the diagonal, followed by any $k \times n$ matrix of orthonormal rows (e.g.\ selection of $k$ coordinates). We provide guarantees (1) and (2) for a larger regime of parameters for which such constructions were previously unknown. Additionally, our construction does not suffer from the extra poly-logarithmic factor multiplying the number of observations $k$ as a function of the sparsity $s$, as present in the currently best known RIP estimates for partial random Fourier matrices and other classes of structured random matrices.

math.NA↗

arXiv2011

Nonuniform Sparse Recovery with Subgaussian Matrices

Compressive sensing predicts that sufficiently sparse vectors can be recovered from highly incomplete information. Efficient recovery methods such as $\ell_1$-minimization find the sparsest solution to certain systems of equations. Random matrices have become a popular choice for the measurement matrix. Indeed, near-optimal uniform recovery results have been shown for such matrices. In this note we focus on nonuniform recovery using Gaussian random matrices and $\ell_1$-minimization. We provide a condition on the number of samples in terms of the sparsity and the signal length which guarantees that a fixed sparse signal can be recovered with a random draw of the matrix using $\ell_1$-minimization. The constant 2 in the condition is optimal, and the proof is rather short compared to a similar result due to Donoho and Tanner.

cs.IT↗

arXiv2011

Low-rank matrix recovery via iteratively reweighted least squares minimization

We present and analyze an efficient implementation of an iteratively reweighted least squares algorithm for recovering a matrix from a small number of linear measurements. The algorithm is designed for the simultaneous promotion of both a minimal nuclear norm and an approximatively low-rank solution. Under the assumption that the linear measurements fulfill a suitable generalization of the Null Space Property known in the context of compressed sensing, the algorithm is guaranteed to recover iteratively any matrix with an error of the order of the best k-rank approximation. In certain relevant cases, for instance for the matrix completion problem, our version of this algorithm can take advantage of the Woodbury matrix identity, which allows to expedite the solution of the least squares problems required at each iteration. We present numerical experiments that confirm the robustness of the algorithm for the solution of matrix completion problems, and demonstrate its competitiveness with respect to other techniques proposed recently in the literature.

math.NA↗

arXiv2011

Sparse Recovery from Combined Fusion Frame Measurements

Sparse representations have emerged as a powerful tool in signal and information processing, culminated by the success of new acquisition and processing techniques such as Compressed Sensing (CS). Fusion frames are very rich new signal representation methods that use collections of subspaces instead of vectors to represent signals. This work combines these exciting 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 does not need to be sparse within each of the subspaces it occupies. This sparsity model is captured using a mixed l1/l2 norm for fusion frames. A signal sparse in a fusion frame can be sampled using very few random projections and exactly reconstructed using a convex optimization that minimizes this mixed l1/l2 norm. The provided sampling conditions generalize coherence and RIP conditions used in standard CS theory. It is demonstrated that they are sufficient to guarantee sparse recovery of any signal sparse in our model. Moreover, a probabilistic analysis is provided using a stochastic model on the sparse signal that shows that under very mild conditions the probability of recovery failure decays exponentially with increasing dimension of the subspaces.

cs.IT↗

arXiv2011

Sparse Legendre expansions via $\ell_1$ minimization

We consider the problem of recovering polynomials that are sparse with respect to the basis of Legendre polynomials from a small number of random samples. In particular, we show that a Legendre s-sparse polynomial of maximal degree N can be recovered from m = O(s log^4(N)) random samples that are chosen independently according to the Chebyshev probability measure. As an efficient recovery method, l1-minimization can be used. We establish these results by verifying the restricted isometry property of a preconditioned random Legendre matrix. We then extend these results to a large class of orthogonal polynomial systems, including the Jacobi polynomials, of which the Legendre polynomials are a special case. Finally, we transpose these results into the setting of approximate recovery for functions in certain infinite-dimensional function spaces.

math.NA↗

arXiv2011

Sparse recovery for spherical harmonic expansions

We show that sparse spherical harmonic expansions can be efficiently recovered from a small number of randomly chosen samples on the sphere. To establish the main result, we verify the restricted isometry property of an associated preconditioned random measurement matrix using recent estimates on the uniform growth of Jacobi polynomials.

math.NA↗

arXiv2010

The Gelfand widths of $\ell_p$-balls for $0<p\leq 1$

We provide sharp lower and upper bounds for the Gelfand widths of $\ell_p$-balls in the $N$-dimensional $\ell_q^N$-space for $0<p\leq 1$ and $p<q \leq 2$. Such estimates are highly relevant to the novel theory of compressive sensing, and our proofs rely on methods from this area.

math.FA↗

arXiv2010

Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type

Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gr{ö}chenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces, and the recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and the first named author that one may replace coherent states related to the group representation by more general abstract continuous frames. In the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces. A unified approach towards atomic decompositions and Banach frames with new results for general coorbit spaces is presented. In the second part we apply the abstract setting to a specific framework and study coorbits of what we call Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel spaces of various types of interest as coorbits. We obtain several old and new wavelet characterizations based on precise smoothness, decay, and vanishing moment assumptions of the respective wavelet. As main examples we obtain results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed smoothness, and even mixtures of the mentioned ones. Due to the generality of our approach, there are many more examples of interest where the abstract coorbit space theory is applicable.

math.FA↗

arXiv2010

Learning Trigonometric Polynomials from Random Samples and Exponential Inequalities for Eigenvalues of Random Matrices

Motivated by problems arising in random sampling of trigonometric polynomials, we derive exponential inequalities for the operator norm of the difference between the sample second moment matrix $n^{-1}U^*U$ and its expectation where $U$ is a complex random $n\times D$ matrix with independent rows. These results immediately imply deviation inequalities for the largest (smallest) eigenvalues of the sample second moment matrix, which in turn lead to results on the condition number of the sample second moment matrix. We also show that trigonometric polynomials in several variables can be learned from $const \cdot D \ln D$ random samples.

math.PR↗

arXiv2010

Compressed Sensing and Redundant Dictionaries

This article 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 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.

math.PR↗

arXiv2010

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling.

cs.IT↗

arXiv2010

Compressive estimation of doubly selective channels in multicarrier systems: Leakage effects and sparsity-enhancing processing

We consider the application of compressed sensing (CS) to the estimation of doubly selective channels within pulse-shaping multicarrier systems (which include 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.

cs.IT↗

arXiv2009

Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation

In this paper, we consider 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. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed l_2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, 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. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.

cs.IT↗

arXiv2009

Circulant and Toeplitz matrices in compressed sensing

Compressed sensing seeks to recover a sparse vector from a small number of linear and non-adaptive measurements. While most work so far focuses on Gaussian or Bernoulli random measurements we investigate the use of partial random circulant and Toeplitz matrices in connection with recovery by $\ell_1$-minization. In contrast to recent work in this direction we allow the use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery result predicts that the necessary number of measurements to ensure sparse reconstruction by $\ell_1$-minimization with random partial circulant or Toeplitz matrices scales linearly in the sparsity up to a $\log$-factor in the ambient dimension. This represents a significant improvement over previous recovery results for such matrices. As a main tool for the proofs we use a new version of the non-commutative Khintchine inequality.

cs.IT↗

arXiv2008

Stability results for random sampling of sparse trigonometric polynomials

Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of non-zero coefficients, can be reconstructed exactly from a small number of random samples using Basis Pursuit (BP) or Orthogonal Matching Pursuit (OMP). In the present article it is shown that recovery by a BP variant is stable under perturbation of the samples values by noise. A similar partial result for OMP is provided. For BP in addition, the stability result is extended to (non-sparse) trigonometric polynomials that can be well-approximated by sparse ones. The theoretical findings are illustrated by numerical experiments.

math.NA↗

arXiv2007

Sparsity in time-frequency representations

We consider signals and operators in finite dimension which have sparse time-frequency representations. As main result we show that an $S$-sparse Gabor representation in $\mathbb{C}^n$ with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that $S\leq Cn/\log(n)$. Our results are applicable to the channel estimation problem in wireless communications and they establish the usefulness of a class of measurement matrices for compressive sensing.

math.CA↗

arXiv2007

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

We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.

math.CA↗

arXiv2006

Recovery algorithms for vector valued data with joint sparsity constraints

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 non-vanishing components. These measures are typically constructed as weighted $\ell_1$ norms of componentwise $\ell_q$ 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 up-dated 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 $\ell_2$-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.

math.NA↗