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Compressed sensing studies linear recovery problems under structure assumptions. We introduce a new class of measurement operators, coined hierarchical measurement operators, and prove results guaranteeing the efficient, stable and robust recovery of hierarchically structured signals from such measurements. We derive bounds on their hierarchical restricted isometry properties based on the restricted isometry constants of their constituent matrices, generalizing and extending prior work on Kronecker-product measurements. As an exemplary application, we apply the theory to two communication scenarios. The fast and scalable HiHTP algorithm is shown to be suitable for solving these types of problems and its performance is evaluated numerically in terms of sparse signal recovery and block detection capability. [ABSTRACT FROM AUTHOR]

The suboptimal performance of wavelets with regard to the approximation of multivariate data gave rise to new representation systems, specifically designed for data with anisotropic features. Some prominent examples of these are given by ridgelets, curvelets, and shearlets, to name a few. The great variety of such so-called directional systems motivated the search for a common framework, which unites many under one roof and enables a simultaneous analysis, for example with respect to approximation properties. Building on the concept of parabolic molecules, the recently introduced framework of α -molecules does in fact include the previous mentioned systems. Until now however it is confined to the bivariate setting, whereas nowadays one often deals with higher dimensional data. This motivates the extension of this unifying theory to dimensions larger than 2, put forward in this work. In particular, we generalize the central result that the cross-Gramian of any two systems of α -molecules will to some extent be localized. As an exemplary application, we investigate the sparse approximation of video signals, which are instances of 3D data. The multivariate theory allows us to derive almost optimal approximation rates for a large class of representation systems. [ABSTRACT FROM AUTHOR]