In this work, we obtain performance guarantees for modified-CS and for its improved
version, modified-CS-Add-LS-Del, for recursive reconstruction of a time sequence of
sparse signals from a reduced set of noisy measurements available at each time. Under
mild assumptions, we show that the support recovery error of both algorithms is bounded
by a time-invariant and small value at all times. The same is also true for the reconstruction
error. Under a slow support change assumption, (i) the support recovery error bound
is small compared to the support size; and (ii) our results hold under weaker assumptions
on the number of measurements than what \(\ell_1\) minimization for noisy data needs.
We first give a general result that only assumes a bound on support size, number of
support changes and number of small magnitude nonzero entries at each time. Later,
we specialize the main idea of these results for two sets of signal change assumptions
that model the class of problems in which a new element that is added to the support
either gets added at a large initial magnitude or its magnitude slowly increases to
a large enough value within a finite delay. Simulation experiments are shown to back
up our claims.