Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices

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

Journal of Statistical Mechanics: Theory and Experiment (2012) P08009

Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make less measurements than what was considered necessary to record a signal, enabling faster or more precise measurement protocols in a wide range of applications. Using an interdisciplinary approach, we have recently proposed in [arXiv:1109.4424] a strategy that allows compressed sensing to be performed at acquisition rates approaching to the theoretical optimal limits. In this paper, we give a more thorough presentation of our approach, and introduce many new results. We present the probabilistic approach to reconstruction and discuss its optimality and robustness. We detail the derivation of the message passing algorithm for reconstruction and expectation max- imization learning of signal-model parameters. We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distribution of signals, and discuss the best possible reconstruction performances regardless of the algorithm. We also present new efficient seeding matrices, test them on synthetic data and analyze their performance asymptotically.

  • 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