Hierarchical Compressed Sensing

Book chapter, 2022

Compressed sensing is a paradigm within signal processing that provides the means for recovering structured signals from linear measurements in a highly efficient manner. Originally devised for the recovery of sparse signals, it has become clear that a similar methodology would also carry over to a wealth of other classes of structured signals. In this work, we provide an overview over the theory of compressed sensing for a particularly rich family of such signals, namely those of hierarchically structured signals. Examples of such signals are constituted by blocked vectors, with only few non-vanishing sparse blocks. We present recovery algorithms based on efficient hierarchical hard thresholding. The algorithms are guaranteed to converge, in a stable fashion with respect to both measurement noise and model mismatches, to the correct solution provided the measurement map acts isometrically restricted to the signal class. We then provide a series of results establishing the required condition for large classes of measurement ensembles. Building upon this machinery, we sketch practical applications of this framework in machine-type communications and quantum tomography.