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Abstract-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.

Abstract-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 ℓ2,1 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.

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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We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank matrix recovery is already well-developed, only few contributions on the low rank tensor recovery problem are available so far. In this paper, we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions, namely the higher order singular value decomposition (HOSVD), the tensor train format (TT), and the general hierarchical Tucker decomposition (HT). We provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank. We show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format. These bounds are extended to partial Fourier maps combined with random sign flips of the tensor entries. Finally, we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from Gaussian random measurements, tensor completion (recovery of missing entries), and Fourier measurements for third order tensors.

We study the recovery of Hermitian low rank matrices X ∈ C n×n from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form a j a * j for some measurement vectors a 1 , . . . , am, i.e., the measurements are given by y j = tr(Xa j a * j ). The case where the matrix X = xx * to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements, y j = | x, a j | 2 via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors a j , j = 1, . . . , m, being chosen independently at random according to a standard Gaussian distribution, or a j being sampled independently from an (approximate) complex projective t-design with t = 4. In the Gaussian case, we require m ≥ Crn measurements, while in the case of 4-designs we need m ≥ Crn log(n). Our results are uniform in the sense that one random choice of the measurement vectors a j guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.Date: October 25, 2014.

A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Gröchenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.