The emergence of compressed sensing has led to renewed interest in the estimation of sparse channels.  Such channels arise in a number of communication environments: terrestrial wireless channels for cellular and vehicular communications, underwater acoustic channels and ultrawideband radio channels.  Most sparse channel estimation methods involve a form of structured channel estimation coupled with a threshold to determine which paths in the channel vector should remain active.  A major contribution of compressed sensing is in the provision of channel estimation cost metrics  based on l_1 and l_0 measures which are the basis of high performance iterative channel estimation algorithms.  Herein, a novel sparse channel estimation algorithm is provided which employs two key features which lead to performance comparable to that of many of the state-of-the-art iterative implementations of compressed sensing, but with lower complexity.  First, an iteration-varying threshold is considered which is coupled with channel updates on subspaces which are “shrinking” with iteration.  The new method is compared with orthogonal matching pursuit, Haupt & Nowak’s scheme for noisy projections, basis pursuit and iterative hard thresholding. Analysis of the new scheme is provided which suggests some clues as to the strong performance.