In this paper we consider the problem of exact recovery of a fixed sparse vector with the measurement matrices sequentially arriving along with corresponding measurements. We propose an extension of the iterative hard thresholding (IHT) algorithm, termed as sequential IHT (SIHT) which breaks the total time horizon into several phases such that IHT is executed in each of these phases using a fixed measurement matrix obtained at the beginning of that phase. We consider a stochastic setting where the measurement matrices obtained at each phase are independent samples of a sub Gaussian random matrix. We prove that if a certain dynamic sample complexity that depends on the sizes of the measurement matrices at each phase, along with their duration and the number of phases, satisfy certain lower bound, the estimation error of SIHT over a fixed time horizon decays rapidly. Interestingly, this bound reveals that the probability of decay of estimation error is hardly affected even if very small number measurements are sporadically used in different phases. This theoretical observation is also corroborated using numerical experiments demonstrating that SIHT enjoys improved probability of recovery compared to offline IHT.

翻译：在本文中,我们考虑了精确回收固定的稀有矢量的问题,测量矩阵依次与相应的测量一起到达。我们建议延长迭代硬阈值算法(IHT),称为相继的IHT(SIHT),将总时间跨度分成几个阶段,使IHT在每一个阶段都使用该阶段开始时获得的固定测量矩阵执行。我们考虑一种随机设置,将每个阶段获得的测量矩阵作为亚高斯随机矩阵的独立样本。我们证明,如果根据每个阶段测量矩阵的大小及其持续时间和数量而确定某种动态的样本复杂性,满足某些较低的约束性,则SIHT在固定时间跨度的估计误差迅速衰减。有趣的是,这一界限表明,即使在不同阶段偶尔使用极小数量的测量,估计误差的可能性也几乎不会受到影响。这种理论观察还用数字实验加以证实,表明SIHT与离线的IHT相比,恢复概率更高。