Sparse stochastic processes: A statistical framework for modern signal processing

Summary form only given. We introduce an extended family of sparse processes that are specified by a generic (non-Gaussian) innovation model or, equivalently, as solutions of linear stochastic differential equations driven by white Lévy noise. We present the mathematical tools for their characterization. The two leading threads of the exposition are - the statistical property of infinite divisibility, which induces two distinct types of behavior - Gaussian vs. sparse at the exclusion of any other; - the structural link between linear stochastic processes and splines. This allows us to prove that these processes admit a parsimonious representation in some matched wavelet-like basis. We show that these models have predictive power for image compression and that they are applicable to the derivation of statistical algorithms for solving ill-posed inverse problems, including compressed sensing.