Data-driven signal decomposition into primary building blocks

Start: 02/08/2016 - 4:15pm
End  : 02/08/2016 - 5:15pm

Applied Math Seminar

Charles K. Chui (Department of Statistics, Stanford University)


The strategy of divide-and-conquer applies to just about all scientific and engineering disciplines for theoretical and algorithmic development as well as experimental implementations for various applications. In mathematics, perhaps one of the most exciting theoretical development in this direction is the notion of “atomic decomposition” for the Hardy spaces $H^p(\R)$ with $0<p\le1$, introduced by Raphy Coifman in a 1974 paper, which contributed to motivate his joint work with Yves Meyer and Elias Stein, published some 10 years later, on the introduction and characterization of the so-called Tent spaces. This significant piece of work has important applications to the unification and simplification of the basic techniques in harmonic analysis. Furthermore, the atomic decomposition of these and other function spaces, contributed by others, has profound impact to the advancement of both harmonic and functional analyses over the decades of the 80’s and 90’s. An important property of Coifman’s atoms for $H^p(\R)$, with $0<p\le1$, is their vanishing moments of order up to $1/p$, leading to the introduction of wavelets and the rapid advances of wavelet analysis and algorithmic development, with applications to most engineering and physical science disciplines for a duration of over two decades.

With the popularity of wavelets, several other families of wavelet-like basis functions were introduced, including: wavelet packets, localized cosines, chirplets, warplets, and mulit-scale Gabor dictionaries. Using this large collection of basis functions as “atoms” to compile a desirable dictionary, the powerful mathematical tool of “basis pursuit”, introduced and studied in some depth by Scott Chen, David Donoho, and Michael Saunders in their popular 1999 SIAM J. Scientific Computing paper, provides a more attractive alternative of the standard iterative computational schemes based on “matching pursuit”. On the other hand, it is very difficult, even if feasible at all, to compile an effective dictionary for sufficiently accurate decomposition of real-world signals in general. In this regard, the basis pursuit approach was recently modified by Thomas Hou and Zuoqiang Shi to “nonlinear basis pursuit”, using a nonlinear optimization scheme. Unfortunately, this approach requires appropriate guesses of initial phase functions, and perhaps the effectiveness of the guess might also depend on the (unknown) number of atoms.

In a recent joint paper with Hrushikesh Mhaskar and my student, Maria van der Walt, of Vanderbilt University, we have initiated a study of the construction of atoms for signal decomposition directly from the input signal itself. The objective of this seminar is to discuss the background of our approach, compare our results with the state-of-the-arts, and conclude the presentation with a brief discussion our computational scheme to signal extrapolation.


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