Stability (over time) of Regularized modified CS(noisy) for recursive causal sparse reconstruction

In this work, we study the “stability” of Regularized modified CS(noisy) for recursive reconstruction of sparse signal sequences from noisy measurements. By “stability” we mean that the number of misses from the current support estimate; the number of extras in it; and the ℓ₂ norm of the reconstruction error remain bounded by a time-invariant value at all times. The concept is meaningful only if the support error bounds are small compared to the signal support size. Regularized modified CS(noisy) is the noisy relaxation of regularized modified CS. The key assumption that reg-mod-CSN uses is that both the sparse signals support and its nonzero signal values change slowly over time. Denote the support estimate from the previous time by T. Modified-CS tries to find a signal that is sparsest outside of T and satisfies the data constraint. Denote the signal estimate from the previous time by μT . Reg-mod-CSN augments mod-CS by also putting the ℓ₂ distance of the current solution from μT as a constraint.