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Sparse Stochastic Processes and Operator-Like Wavelet Expansions

M. Unser

Keynote, Proceedings of the Fifth International Conference on Computational Harmonic Analysis, Twenty-Ninth Annual Shanks Lecture (CHA'14), Vanderbilt TN, USA, May 19-23, 2014, pp. 61.



We introduce an extended family of continuous-domain sparse processes that are specified by a generic (non-Gaussian) innovation model or, equivalently, as solutions of linear stochastic differential equations driven by white Lvy noise. We present the functional tools for their characterization. We show that their probability distributions are infinitely divisible, which induces two distinct types of behaviorGaussian vs. sparseat the exclusion of any other. This is the key to proving that the non-Gaussian members of the family admit a sparse representation in a matched wavelet basis.

We use the characteristic form of these processes to deduce their transform-domain statistics and to precisely assess residual dependencies. These ideas are illustrated with examples of sparse processes for which operator-like wavelets outperform the classical KLT (or DCT) and result in an independent component analysis. Finally, for the case of self-similar processes, we show that the wavelet-domain probability laws are ruled by a diffusion-like equation that describes their evolution across scale.


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