Sparse Approximation and Atomic Decomposition: Considering Atom Interactions in Evaluating and Building Signal Representations

Dr. Bob Sturm
UC Santa Barbara

Thursday, February 19, 2009 at 1:00pm
Engr IV Room 53-135

Abstract
I will present work from my recent dissertation, which makes contributions to the sparse approximation and efficient representation of complex signals, e.g., acoustic signals, using greedy iterative descent pursuits and overcomplete dictionaries. As others have noted before, peculiar problems arise when a signal model is mismatched to the signal content, and a pursuit makes bad selections from the dictionary. These result in models that contain several atoms having no physical significance to the signal, and instead exist to correct the representation through destructive interference. This diminishes the efficiency of the generated signal model, and hinder the useful application of sparse approximation to signal analysis (e.g., source identification), visualization (e.g., source selection), and modification (e.g., source extraction). While past works have addressed these problems by reformulating a pursuit to avoid them, in this dissertation we use these corrective terms to learn about the signal, the pursuit algorithm, the dictionary, and the created model. Our thesis is essentially that a better signal model results when a pursuit builds it considering the interaction between the atoms. We formally study these effects and propose novel measures of them to quantify the interaction between atoms in a model, and to illuminate the role of each atom in representing a signal. We propose and study different ways of incorporating these new measures into the atom selection criteria of greedy iterative descent pursuits, and show analytically and empirically that these interference-adaptive pursuits can produce models with increased efficiency and meaningfulness.

Biography
Dr. Sturm has received an undergraduate degree in physics from the University of Colorado, Boulder (B.A. 1998), a graduate degree in computer music from Stanford University (M.A. 1999), and a few other graduate degrees from the University of California, Santa Barbara (M.S. 2004, M.S. 2007, Ph.D. 2009). He will continue his research this March in sparse approximation and signal representation as a Chateaubriand Fellow post-doctoral researcher at the Université Pierre et Marie Curie, Paris 6.