Celebratio Mathematica — Daubechies

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I. Daubechies, M. Defrise, and C. De Mol: “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. 57 : 11 (2004), pp. 1413–1457. MR 2077704 Zbl 1077.65055 ArXiv math/0307152 article

We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted \( \ell^p \)-penalties on the coefficients of such expansions, with \( 1\leq p \) \( \leq 2 \), still regularizes the problem. Use of such \( \ell^p \)-penalized problems with \( p < 2 \) is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm.

Christine De Mol	Related
Michel Defrise	Related

I. Daubechies, M. Defrise, and C. De Mol: “Sparsity-enforcing regularisation and ISTA revisited,” Inverse Problems 32 : 10 (2016). Article ID 104001, 15 pages. MR 3627017 Zbl 06652332 article

About two decades ago, the concept of sparsity emerged in different disciplines such as statistics, imaging, signal processing and inverse problems, and proved to be useful for several applications. Sparsity-enforcing constraints or penalties were then shown to provide a viable alternative to the usual quadratic ones for the regularisation of ill-posed problems. To compute the corresponding regularised solutions, a simple, iterative and provably convergent algorithm was proposed and later on referred to as the iterative soft-thresholding algorithm. This paper provides a brief review of these early results as well as that of the subsequent literature, albeit from the authors’ limited perspective. It also presents the previously unpublished proof of an extension of the original framework.

Christine De Mol	Related
Michel Defrise	Related