Stability (over time) of Modified-CS and LS-CS for Recursive Causal Sparse Reconstruction

In this work, we obtain sufficient conditions for the “stability” of our recently proposed algorithms, modified-CS (for noisy measurements) and Least Squares CS-residual (LS-CS), designed for recursive reconstruction of sparse signal sequences from noisy measurements. By “stability” we mean that the number of misses from the current support estimate and the number of extras in it remain bounded by a time-invariant value at all times. The concept is meaningful only if the bound is small compared to the current signal support size. A direct corollary is that the reconstruction errors are also bounded by a time-invariant and small value.