Statistical physics-based reconstruction in compressed sensing

Florent Krzakala 1, Marc Mézard 2, François Sausset 2, Yifan Sun 1, 3, Lenka Zdeborová 4

Physical Review X 2 (2012) 021005

Compressed sensing is triggering a major evolution in signal acquisition. It consists in sampling a sparse signal at low rate and later using computational power for its exact reconstruction, so that only the necessary information is measured. Currently used reconstruction techniques are, however, limited to acquisition rates larger than the true density of the signal. We design a new procedure which is able to reconstruct exactly the signal with a number of measurements that approaches the theoretical limit in the limit of large systems. It is based on the joint use of three essential ingredients: a probabilistic approach to signal reconstruction, a message-passing algorithm adapted from belief propagation, and a careful design of the measurement matrix inspired from the theory of crystal nucleation. The performance of this new algorithm is analyzed by statistical physics methods. The obtained improvement is confirmed by numerical studies of several cases.

  • 1. Laboratoire de Physico-Chimie Théorique (LPCT),
    CNRS : UMR7083 – ESPCI ParisTech
  • 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 3. LMIB and School of Mathematics and Systems Science,,
    Beihang University
  • 4. Institut de Physique Théorique (ex SPhT) (IPHT),
    CNRS : URA2306 – CEA : DSM/IPHT