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Hierarchical Compressed Sensing

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Compressed Sensing in Information Processing

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

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.

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Acknowledgements

This work is a report of some of the findings of the DFG-funded project EI 519/9-1 within the priority programme ‘Compressed Sensing in Information Processing’ (CoSIP), jointly held by J. Eisert and G. Wunder. We specifically thank our coauthors, in particular, M. Barzegar, G. Caire, R. Fritscheck, S. Haghighatshoar, D. Hangleiter, M. Kliesch, S. Stefanatos, R. Kueng, and J. Wilkens, with which we have explored this research theme over the years.

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Authors and Affiliations

  1. Dahlem Center for Complex Quantum Systems and Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany

    Jens Eisert

  2. Institute for Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden

    Axel Flinth

  3. Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany

    Benedikt Groß & Gerhard Wunder

  4. Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE

    Ingo Roth

  5. Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany

    Ingo Roth

Authors
  1. Jens Eisert
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  2. Axel Flinth
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  3. Benedikt Groß
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  4. Ingo Roth
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  5. Gerhard Wunder
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Corresponding author

Correspondence to Gerhard Wunder .

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Editors and Affiliations

  1. Mathematisches Institut, Ludwig Maximilian University of Munich, München, Bayern, Germany

    Gitta Kutyniok

  2. Lehrstuhl für Mathematik, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Holger Rauhut

  3. Lehrstuhl für Mathematik, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Robert J. Kunsch

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Eisert, J., Flinth, A., Groß, B., Roth, I., Wunder, G. (2022). Hierarchical Compressed Sensing. In: Kutyniok, G., Rauhut, H., Kunsch, R.J. (eds) Compressed Sensing in Information Processing. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-09745-4_1

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