Streaming measurements in compressive sensing: ℓ1 filtering

The central framework for signal recovery in compressive sensing is lscr₁ norm minimization. In recent years, tremendous progress has been made on algorithms, typically based on some kind of gradient descent or Newton iterations, for performing lscr₁ norm minimization. These algorithms, however, are for the most part ldquostaticrdquo: they focus on finding the solution for a fixed set of measurements. In this paper, we will present a method for quickly updating the solution to some lscr₁ norm minimization problems as new measurements are added. The result is an ldquolscr₁ filterrdquo and can be implemented using standard techniques from numerical linear algebra. Our proposed scheme is homotopy based where we add new measurements in the system and instead of solving updated problem directly, we solve a series of simple (easy to solve) intermediate problems which lead to the desired solution.