Quantized Compressed Sensing by Rectified Linear Units

PAMM, vol. 20, pp. 4125–4149

Abstract

Abstract This work is concerned with the problem of recovering high‐dimensional signals, which belong to a convex set of low‐complexity, from a small number of quantized measurements. We propose to estimate the signals via a convex program based on rectified linear units (ReLUs) for two different quantization schemes, namely one‐bit and uniform multi‐bit quantization. Assuming that the linear measurement process can be modelled by a sensing matrix with i.i.d. subgaussian rows, we obtain for both schemes near‐optimal uniform reconstruction guarantees by adding well‐designed noise to the linear measurements prior to the quantization step. In the one‐bit case, we show that the program is robust against adversarial bit corruptions as well as additive noise on the linear measurements. Further, our analysis quantifies precisely how the rate‐distortion relationship of the program changes depending on whether we seek reconstruction accuracies above or below the noise floor. The proofs rely on recent results by Dirksen and Mendelson on non‐Gaussian hyperplane tessellations.

Authors 4

1. Hans Christian Jung Aachen Lehrstuhl für Mathematik der Informationsverarbeitung

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Mathematik der Informationsverarbeitung RWTH Aachen Pontdriesch 10 DE-52062 Aachen

   Lehrstuhl für Mathematik der Informationsverarbeitung RWTH Aachen Pontdriesch 10 DE-52062Aachen

2. Johannes Maly corresponding Aachen Lehrstuhl für Mathematik der Informationsverarbeitung

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Mathematik der Informationsverarbeitung RWTH Aachen Pontdriesch 10 DE-52062 Aachen

   Lehrstuhl für Mathematik der Informationsverarbeitung RWTH Aachen Pontdriesch 10 DE-52062Aachen

3. Lars Palzer

   Technical University of Munich

   Affiliation as printed

   Lehrstuhl für Nachrichtentechnik TU München Theresienstraße 90 DE-80333 München

   Lehrstuhl für Nachrichtentechnik TU München Theresienstraße 90 DE-80333München

4. Alexander Stollenwerk

   Technische Universität Berlin

   Affiliation as printed

   Institut für Mathematik TU Berlin Straße des 17. Juni 136 DE-10623 Berlin

   Institut für Mathematik TU Berlin Straße des 17. Juni 136 DE-10623Berlin