Shrinkage Rules for Variational Minimization Problems and Applications to Analytical Ultracentrifugation

TL;DR

This paper introduces fast, q-dependent shrinkage rules for variational minimization problems with l_q-sparsity constraints, enabling efficient solutions for noisy signal representation and applications like ultracentrifugation.

Contribution

It develops new q-adapted shrinkage rules that approximate minimizers efficiently, with applications demonstrated in ultracentrifugation data analysis.

Findings

Proposed shrinkage rules are computationally fast and effective.

Application yields sparser, sharper solutions with higher resolution.

Outperforms standard regularization in ultracentrifugation tasks.

Abstract

Finding a sparse representation of a possibly noisy signal can be modeled as a variational minimization with l_q-sparsity constraints for q less than one. Especially for real-time, on-line, or iterative applications, in which problems of this type have to be solved multiple times, one needs fast algorithms to compute these minimizers. 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.