Linear convergence of stochastic iterative greedy algorithms with sparse constraints

Motivated by recent work on stochastic gradient descent methods, we develop two stochastic variants of greedy algorithms for possibly non-convex optimization problems with sparsity constraints. We prove linear convergence1 in expectation to the solution within a specified tolerance. This generalized framework is specialized to the problems of sparse signal recovery in compressed sensing and low-rank matrix recovery, giving methods with provable convergence guarantees that often outperform their deterministic counterparts. We also analyze the settings, where gradients and projections can only be computed approximately, and prove the methods are robust to these approximations. We include many numerical experiments, which align with the theoretical analysis and demonstrate these improvements in several different settings.