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An Introduction to Sparse Stochastic Processes

M. Unser

Tutorial, Thirty-Ninth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'14), Firenze, Italian Republic, May 4-9, 2014.

Sparse stochastic processes are continuous-domain processes that admit a parsimonious representation in some matched wavelet-like basis. Such models are relevant for image compression, compressed sensing, and, more generally, for the derivation of statistical algorithms for solving ill-posed inverse problems.

This tutorial provides an introduction to the 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 provide a complete functional characterization of these processes and highlight some of their properties. The two leading threads that underlie the exposition are:

  1. the statistical property of infinite divisibility, which induces two distinct types of behavior—Gaussian vs. sparse—at the exclusion of any other;
  2. the structural link between linear stochastic processes and splines.

The formalism lends itself to the derivation of the transform-domain statistics of these processes and to the identification of "optimal" (ICA-like) representations. We also show that these models are applicable to the derivation of statistical algorithms for solving ill-posed inverse problems, including compressed sensing. The proposed formulation leads to a reinterpretation of popular sparsity-promoting processing schemes—such as total-variation denoising, LASSO, and wavelet shrinkage—as MAP estimators for specific types of sparse processes, but it also suggests alternative Bayesian recovery procedures that minimize the estimation error.

The lecture notes for the tutorial are available on the web at http://www.sparseprocesses.org/ 

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