  Compressed sensing allows perfect recovery of sparse signals (or signals
sparse in some basis) using only a small number of random measurements.
Existing results in compressed sensing literature have focused on
characterizing the achievable performance by bounding the number of samples
required for a given level of signal sparsity. However, using these bounds to
minimize the number of samples requires a-priori knowledge of the sparsity of
the unknown signal, or the decay structure for near-sparse signals.
Furthermore, there are some popular recovery methods for which no such bounds
are known.
  In this paper, we investigate an alternative scenario where observations are
available in sequence. For any recovery method, this means that there is now a
sequence of candidate reconstructions. We propose a method to estimate the
reconstruction error directly from the samples themselves, for every candidate
in this sequence. This estimate is universal in the sense that it is based only
on the measurement ensemble, and not on the recovery method or any assumed
level of sparsity of the unknown signal. With these estimates, one can now stop
observations as soon as there is reasonable certainty of either exact or
sufficiently accurate reconstruction. They also provide a way to obtain
"run-time" guarantees for recovery methods that otherwise lack a-priori
performance bounds.
  We investigate both continuous (e.g. Gaussian) and discrete (e.g. Bernoulli)
random measurement ensembles, both for exactly sparse and general near-sparse
signals, and with both noisy and noiseless measurements.
