On the Effective Sample Complexity for Exact Sparse Recovery from Sequential Linear Measurements

In this paper we consider the problem of exact recovery of a fixed sparse vector from sequentially arriving measurements. We assume that the measurements are generated by a linear model with time varying matrices and both the measurement vector as well as the matrix at each time are made available. However, we assume that the underlying unknown sparse vector is fixed during the time of interest. We prove that if the measurement matrices are i.i.d. subGaussian, the iterates produced by the popular iterative hard thresholding (IHT) algorithm can converge to the exact sparse vector with high probability if a certain function of the sample complexities of the time varying measurements, which we call effective sample complexity satisfies certain lower bound dependent on K,N, the sparsity and the length of the unknown vector, respectively. Interestingly, this bound reveals that the probability that the estimation error at the end of some instant is small enough, is hardly affected even if very small number measurements are used at sporadically chosen time instances. We also corroborate this theoretical result with numerical experiments which demonstrate that the conventional IHT can enjoy greater probability of recovery by occasionally using far lesser number of measurements than that required for successful recovery with offline IHT with fixed measurement matrix.