Compressed sensing for finite-valued signals

Keiper, Sandra ; Kutyniok, Gitta ; Lee, Dae Gwan und Pfander, Götz E. (1. November 2017): Compressed sensing for finite-valued signals. In: Linear Algebra and its Applications, Bd. 532: S. 570-613

Abstract

The need of reconstructing discrete-valued sparse signals from few measurements, that is solving an undetermined system of linear equations, appears frequently in science and engineering. Whereas classical compressed sensing algorithms do not incorporate the additional knowledge of the discrete nature of the signal, classical lattice decoding approaches such as the sphere decoder do not utilize sparsity constraints.

In this work, we present an approach that incorporates a discrete values prior into basis pursuit. In particular, we address unipolar binary and bipolar ternary sparse signals, i.e., sparse signals with entries in {0, 1}, respectively in {−1, 0, 1}. We will show that phase transition takes place earlier than when using the classical basis pursuit approach and that, independently of the sparsity of the signal, at most N/2, respectively 3N/4, measurements are necessary to recover a unipolar binary, and a bipolar ternary signal uniquely, where N is the dimension of the ambient space. We will further discuss robustness of the algorithm and generalizations to signals with entries in larger alphabets.

• Dokumententyp: Zeitschriftenartikel
• Keywords: Compressed Sensing; Sparse Recovery; Null Space Property; Finite Alphabet; Statistical Dimension; Phase Transitions
• Fakultät: Mathematik, Informatik und Statistik > Mathematik > Professur für Mathematische Grundlagen des Verständnisses der künstlichen Intelligenz
• Themengebiete: 500 Naturwissenschaften und Mathematik > 510 Mathematik
• ISSN: 00243795
• Sprache: Englisch