  Despite the importance of sparsity signal models and the increasing
prevalence of high-dimensional streaming data, there are relatively few
algorithms for dynamic filtering of time-varying sparse signals. Of the
existing algorithms, fewer still provide strong performance guarantees. This
paper examines two algorithms for dynamic filtering of sparse signals that are
based on efficient l1 optimization methods. We first present an analysis for
one simple algorithm (BPDN-DF) that works well when the system dynamics are
known exactly. We then introduce a novel second algorithm (RWL1-DF) that is
more computationally complex than BPDN-DF but performs better in practice,
especially in the case where the system dynamics model is inaccurate.
Robustness to model inaccuracy is achieved by using a hierarchical
probabilistic data model and propagating higher-order statistics from the
previous estimate (akin to Kalman filtering) in the sparse inference process.
We demonstrate the properties of these algorithms on both simulated data as
well as natural video sequences. Taken together, the algorithms presented in
this paper represent the first strong performance analysis of dynamic filtering
algorithms for time-varying sparse signals as well as state-of-the-art
performance in this emerging application.
