Iterative Hard Thresholding for Weighted Sparse Approximation

Recent work by Rauhut and Ward developed a notion of weighted sparsity and a corresponding notion of Restricted Isometry Property for the space of weighted sparse signals. Using these notions, we pose a best weighted sparse approximation problem, i.e. we seek structured sparse solutions to underdetermined systems of linear equations. Many computationally efficient greedy algorithms have been developed to solve the problem of best $s$-sparse approximation. The design of all of these algorithms employ a similar template of exploiting the RIP and computing projections onto the space of sparse vectors. We present an extension of the Iterative Hard Thresholding (IHT) algorithm to solve the weighted sparse approximation problem. This IHT extension employs a weighted analogue of the template employed by all greedy sparse approximation algorithms. Theoretical guarantees are presented and much of the original analysis remains unchanged and extends quite naturally. However, not all the theoretical analysis extends. To this end, we identify and discuss the barrier to extension. Much like IHT, our IHT extension requires computing a projection onto a non-convex space. However unlike IHT and other greedy methods which deal with the classical notion of sparsity, no simple method is known for computing projections onto these weighted sparse spaces. Therefore we employ a surrogate for the projection and analyze its empirical performance.