  Compressed sensing typically deals with the estimation of a system input from
its noise-corrupted linear measurements, where the number of measurements is
smaller than the number of input components. The performance of the estimation
process is usually quantified by some standard error metric such as squared
error or support set error. In this correspondence, we consider a noisy
compressed sensing problem with any arbitrary error metric. We propose a
simple, fast, and highly general algorithm that estimates the original signal
by minimizing the error metric defined by the user. We verify that our
algorithm is optimal owing to the decoupling principle, and we describe a
general method to compute the fundamental information-theoretic performance
limit for any error metric. We provide two example metrics --- minimum mean
absolute error and minimum mean support error --- and give the theoretical
performance limits for these two cases. Experimental results show that our
algorithm outperforms methods such as relaxed belief propagation (relaxed BP)
and compressive sampling matching pursuit (CoSaMP), and reaches the suggested
theoretical limits for our two example metrics.
