Celebratio Mathematica

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

J. Brodie, I. Daubechies, C. De Mol, D. Giannone, and I. Loris: “Sparse and stable Markowitz portfolios,” Proc. Natl. Acad. Sci. USA 106 : 30 (2009), pp. 12267–12272. Zbl 1203.91271 ArXiv 0708.0046 article

We consider the problem of portfolio selection within the classical Markowitz mean-variance framework, reformulated as a constrained least-squares regression problem. We propose to add to the objective function a penalty proportional to the sum of the absolute values of the portfolio weights. This penalty regularizes (stabilizes) the optimization problem, encourages sparse portfolios (i.e., portfolios with only few active positions), and allows accounting for transaction costs. Our approach recovers as special cases the no-short-positions portfolios, but does allow for short positions in limited number. We implement this methodology on two benchmark data sets constructed by Fama and French. Using only a modest amount of training data, we construct portfolios whose out-of-sample performance, as measured by Sharpe ratio, is consistently and significantly better than that of the naïve evenly weighted portfolio.

Ignace Loris	Related
Christine De Mol	Related
Domenico Giannone	Related
Joshua Brodie	Related

L. Jacques, C. De Vleeschouwer, Y. Boursier, P. Sudhakar, C. De Mol, A. Pizurica, S. Anthoine, P. Vandergheynst, P. Frossard, C. Bilen, S. Kitic, N. Bertin, R. Gribonval, N. Boumal, B. Mishra, P.-A. Absil, R. Sepulchre, S. Bundervoet, C. Schretter, A. Dooms, P. Schelkens, O. Chabiron, F. Malgouyres, J.-Y. Tourneret, P. Dobigeon, N. Chainais, C. Richard, B. Cornelis, I. Daubechies, D. Dunson, M. Dankova, P. Rajmic, K. Degraux, V. Cambareri, B. Geelen, G. Lafruit, G. Setti, J.-F. Determe, J. Louveaux, F. Horlin, A. Drémeau, P. Heas, C. Herzet, V. Duval, G. Peyré, A. Fawzi, M. Davies, N. Gillis, S. A. Vavasis, C. Soussen, L. Le Magoarou, J. Liang, J. Fadili, A. Liutkus, D. Martina, S. Gigan, L. Daudet, M. Maggioni, S. Minsker, N. Strawn, C. Mory, F. Ngole, J.-L. Starck, I. Loris, S. Vaiter, M. Golbabaee, and D. Vukobratovic: Proceedings of the second “international Traveling Workshop on Interactions between Sparse models and Technology” (iTWIST’14). Preprint, October 2014. Namur, Belgium, 27–29 August 2014. techreport

The implicit objective of the biennial “international–Traveling Workshop on Interactions between Sparse models and Technology” (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. iTWIST’14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the “sparsity paradigm”: Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What’s next?; Sparse machine learning and inference.

Angélique Drémeau	Related
Fred Maurice Ngolè Mboula	Related
Vincent Duval	Related
Gabriel Peyré	Related
Aleksandra Pižurica	Related
Jalal M. Fadili	Related
Alhussein Fawzi	Related
Pavel Rajmic	Related
Pascal Frossard	Related
Bert Geelen	Related
Cédric Richard	Related
Sylvain Gigan	Related
Nicolas Gillis	Related
Mohammad Golbabaee	Related
Rémi Gribonval	Related
Peter Schelkens	Related
Colas Schretter	Related
Rodolphe J. Sepulchre	Related
Gianluca Setti	Related
Patrick Héas	Related
Cédric Herzet	Related
François Horlin	Related
Charles Soussen	Related
Jean-Luc Starck	Related
Pierre-Antoine Absil	Related
Laurent Jacques	Related
Nathaniel Strawn	Related
Prasad Sudhakar	Related
Sandrine Anthoine	Related
Jean-Yves Tourneret	Related
Nancy Bertin	Related
Srđan Kitić	Related
Samuel Vaiter	Related
Çağdaş Bilen	Related
Pierre Vandergheynst	Related
Stephen Andrew Vavasis	Related
Nicolas Boumal	Related
Yannick Boursier	Related
Christophe De Vleeschouwer	Related
Gauthier Lafruit	Related
Dejan Vukobratović	Related
Shaun Bundervoet	Related
Jingwei Liang	Related
Valerio Cambareri	Related
Antoine Liutkus	Related
Ignace Loris	Related
Olivier Chabiron	Related
Bruno Cornelis	Related
Pierre Chainais	Related
David Brian Dunson	Related
Jérôme Louveaux	Related
Mauro Maggioni	Related
Luc Le Magoarou	Related
François Malgouyres	Related
David Martina	Related
Stanislav Minsker	Related
Bandev Mishra	Related
Marie Daňková	Related
Laurent Daudet	Related
Michael Evan Davies	Related
Christine De Mol	Related
Kévin Degraux	Related
Cyril Mory	Related
J.-F. Determe	Related
Nicolas Dobigeon	Related
Ann Dooms	Related

@techreport {key89916570,
	AUTHOR = {Jacques, L. and De Vleeschouwer, C.
		 and Boursier, Y. and Sudhakar, P. and
		 De Mol, C. and Pizurica, A. and Anthoine,
		 S. and Vandergheynst, P. and Frossard,
		 P. and Bilen, C. and Kitic, S. and Bertin,
		 N. and Gribonval, R. and Boumal, N.
		 and Mishra, B. and Absil, P.-A. and
		 Sepulchre, R. and Bundervoet, S. and
		 Schretter, C. and Dooms, A. and Schelkens,
		 P. and Chabiron, O. and Malgouyres,
		 F. and Tourneret, J.-Y. and Dobigeon,
		 P. and Chainais, N. and Richard, C.
		 and Cornelis, B. and Daubechies, I.
		 and Dunson, D. and Dankova, M. and Rajmic,
		 P. and Degraux, K. and Cambareri, V.
		 and Geelen, B. and Lafruit, G. and Setti,
		 G. and Determe, J.-F. and Louveaux,
		 J. and Horlin, F. and Dr\'emeau, A.
		 and Heas, P. and Herzet, C. and Duval,
		 V. and Peyr\'e, G. and Fawzi, A. and
		 Davies, M. and Gillis, N. and Vavasis,
		 S. A. and Soussen, C. and Le Magoarou,
		 L. and Liang, J. and Fadili, J. and
		 Liutkus, A. and Martina, D. and Gigan,
		 S. and Daudet, L. and Maggioni, M. and
		 Minsker, S. and Strawn, N. and Mory,
		 C. and Ngole, F. and Starck, J.-L. and
		 Loris, I. and Vaiter, S. and Golbabaee,
		 M. and Vukobratovic, D.},
	TITLE = {Proceedings of the second ``international
		 {T}raveling {W}orkshop on {I}nteractions
		 between {S}parse models and {T}echnology''
		 (i{TWIST}'14)},
	TYPE = {Preprint},
	MONTH = {October},
	YEAR = {2014},
	PAGES = {69},
	NOTE = {Namur, Belgium, 27--29 August 2014.},
}

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

year	title	people

			clear

Ingrid Daubechies

Complete Bibliography

Works connected to Christine De Mol

Filter the Bibliography List