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Analyzing Least Squares and Kalman Filtered Compressed Sensing

In recent work, we studied the problem of causally reconstructing time sequences of spatially sparse signals, with unknown and slow time-varying sparsity patterns, from a limited number of linear "incoherent" measurements. We proposed a solution called Kalman Filtered Compressed Sensing (KF-CS). The key idea is to run a reduced order KF only for the current signal's estimated nonzero coefficients' set, while performing CS on the Kalman filtering error to estimate new additions, if any, to the set. KF may be replaced by Least Squares (LS) estimation and we call the resulting algorithm LS-CS. In this work, (a) we bound the error in performing CS on the LS error and (b) we obtain the conditions under which the KF-CS (or LS-CS) estimate converges to that of a genie-aided KF (or LS), i.e. the KF (or LS) which knows the true nonzero sets.