Nonzero elements of a signal vector, which may be a sparse signal vector, may be determined based on an observation vector representing a set of underdetermined observations using a compressed sensing optimization and a non-underdetermined estimation method such as iterative linear minimum mean-square error (“LMMSE”) estimation. Compressed sensing optimization may be used to obtain a subset of potentially nonzero elements of the signal vector, and LMMSE estimation may then be used to find the nonzero elements among the potentially nonzero elements. The identification of nonzero elements may then be used to recover the signal vector from the observation vector. This technique is useful for recovering compressed data such as a sparse frequency space representation of audio or video data from a measurement. The technique is also useful for identifying at a base station a relatively small number active devices in an overloaded communication network.