Statistical compressive sensing of Gaussian mixture models

A new framework of compressive sensing (CS), namely statistical compressive sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution and achieving accurate reconstruction on average, is introduced. For signals following a Gaussian distribution, with Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably smaller than the O(k log(N/k)) required by conventional CS, where N is the signal dimension, and with an optimal decoder implemented with linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the best k-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best k-term approximation with probability one, and the bound constant can be efficiently calculated. For signals following Gaussian mixture models, SCS with a piecewise linear decoder is introduced and shown to produce for real images better results than conventional CS based on sparse models.