Convergence of Basis Pursuit De-noising with Dynamic Filtering

Causal inference of dynamically changing signals is a vital task in many applications, including real-time image processing and channel estimation. Over the past few years, many algorithms have been proposed to accomplish this task, but extremely few algorithms have any theoretical guarantees on stability, convergence or performance. In this work we use results from the sparsity-based signal processing literature to demonstrate some basic bounds for one particular algorithm: basis pursuit de-noising with dynamic filtering (BPDN-DF). We show for what parameter ranges the algorithm remains stable for, and provide some guarantees on the steady-state approximation error.