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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Shrinkage rules for variational minimization problems and applications to analytical ultracentrifugation

1Helmholtz Zentrum München, Institute of Biomathematics and Biometry, Ingolstädter Landstrasse 1, 85764 Neuherberg, Germany.

2Section on Medical Biophysics, Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, 9 Memorial Drive, Bethesda, MD 20892, USA.

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 19, Issue 4-5, Pages 593–614, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/jiip.2011.057, November 2011

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Finding a sparse representation of a noisy signal can be modeled as a variational minimization with -sparsity constraints for q less than one. Especially for real-time, online, or iterative applications, in which problems of this type have to be solved multiple times, one needs fast algorithms to compute these minimizers. However, identifying the exact minimizers is computationally expensive. We consider minimization up to a constant factor to circumvent this limitation. We verify that q-dependent modifications of shrinkage rules provide closed formulas for such minimizers. Therefore, their computation is extremely fast. We also introduce a new shrinkage rule which is adapted to q. To support the theoretical results, the proposed method is applied to Landweber iteration with shrinkage used at each iteration step. This approach is utilized to solve the ill-posed problem of analytic ultracentrifugation, a method to determine the size distribution of macromolecules. For relatively pure solutes, our proposed scheme leads to sparser solutions with sharper peaks, higher resolution, and smaller residuals than standard regularization for this problem.

Keywords.: Shrinkage; variational optimization; sparsity; frames; Fredholm integral equations

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