Stochastic Systems Group  

Compressed Sensing with Sequential Observations
Dmitry M. Malioutov
SSG, MIT
Compressed sensing allows perfect recovery of a signal that is known to be sparse in some basis using only a small number of measurements. The results in the literature have focused on the asymptotics of how many samples are required and the probability of making an error for a fixed batch of samples. We investigate an alternative scenario where observations are available in sequence and can be stopped as soon as there is reasonable certainty of correct reconstruction. For the random Gaussian ensemble we show that a simple stopping rule gives the absolute minimum number of observations required for exact recovery, with probability one. However, for other ensembles like Bernoulli or Fourier, this is no longer true, and the rule is modified to trade off delay in stopping and probability of error. We also describe a stopping rule for the nearsparse case which tells when enough observations are made to reach a desired tolerance in reconstruction. Sequential approach to compressed sensing involves the solution of a sequence of linear programs, and we outline how this sequence can be solved efficiently.
Joint work with Sujay Sanghavi and Alan Willsky.
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