  We study the problem of recursively recovering a time sequence of sparse
vectors, St, from measurements Mt := St + Lt that are corrupted by structured
noise Lt which is dense and can have large magnitude. The structure that we
require is that Lt should lie in a low dimensional subspace that is either
fixed or changes "slowly enough"; and the eigenvalues of its covariance matrix
are "clustered". We do not assume any model on the sequence of sparse vectors.
Their support sets and their nonzero element values may be either independent
or correlated over time (usually in many applications they are correlated). The
only thing required is that there be some support change every so often. We
introduce a novel solution approach called Recursive Projected Compressive
Sensing with cluster-PCA (ReProCS-cPCA) that addresses some of the limitations
of earlier work. Under mild assumptions, we show that, with high probability,
ReProCS-cPCA can exactly recover the support set of St at all times; and the
reconstruction errors of both St and Lt are upper bounded by a time-invariant
and small value.
