  Compressed Sensing algorithms often make use of the hard thresholding
operator to pass from dense vectors to their best s-sparse approximations.
However, the output of the hard thresholding operator does not depend on any
information from a particular problem instance. We propose an alternative
thresholding rule, Look Ahead Thresholding, that does. In this paper we offer
both theoretical and experimental justification for the use of this new
thresholding rule throughout compressed sensing.
