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- Holger Rauhut
- 2009

These notes give a mathematical introduction to compressive sensing focusing on recovery using 1-minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate… (More)

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… (More)

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… (More)

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… (More)

We investigate the problem of reconstructing sparse multivariate trigonometric polyno-mials 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… (More)

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… (More)

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… (More)

- Massimo Fornasierand, Holger Rauhut
- 2007

This article provides a variational formulation for hard and firm thresholding. A related functional can be used to regularize inverse problems by sparsity constraints. We show that a damped hard or firm thresholded Landweber iteration converges to its minimizer. This provides an alternative to an algorithm recently studied by the authors. We prove… (More)

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… (More)