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Towards a Theory of Sparse Stochastic Processes, or when Paul Lévy Joins Forces with Nobert Wiener

M. Unser

Mathematics and Image Analysis 2012 (MIA'12), Paris, French Republic, January 16-18, 2012.

The current formulations of compressed sensing and sparse signal recovery are based on solid variational principles, but they are fundamentally deterministic. By drawing on the analogy with the classical theory of signal processing, it is likely that further progress may be achieved by adopting a statistical (or estimation theoretic) point of view. Here, we shall argue that Paul Lévy (1886-1971), who was one of the very early proponents of Haar wavelets, was in advance over his time, once more. He is the originator of the Lévy-Khinchine formula, which happens to be the perfect (non-Gaussian) ingredient to support a continuous-domain theory of sparse stochastic processes. 

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