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Constructions déterministes pour la régression parcimonieuse

Abstract : In this thesis we investigate some deterministic designs for the sparse regression. Our issue is mainly inspired by Compressed Sensing which is concerned by simultaneously acquire and compress a signal of large size from a small number of linear measurements. More precisely, we show that there exists a link between variable selection and prediction error with standard estimators (such as the lasso, the Dantzig selector, the basis pursuit) and the distortion, which measures how "far" is the Manhattan norm from the Euclidean norm, of the null-space of the design. Hence, we show that every construction of subspaces with low-distortion (called "almost"-Euclidean subspaces) gives "good" designs. In a second part, we are interested by designs constructed from unbalanced expander graphs. We accurately established their performances in terms of variable selection and prediction error. Finally, we are interested in the faithful reconstruction of signed measures on the real line. We show that every generalized Vandermonde system gives design such one can exactly recover all the sparse vectors from a dramatically small number of observations. In an independent part, we investigate the stability of the isoperimetric inequalities for the log-concave measures on the real line.
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Contributor : Yohann de Castro <>
Submitted on : Wednesday, January 4, 2012 - 12:56:06 PM
Last modification on : Thursday, March 5, 2020 - 5:57:10 PM
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  • HAL Id : tel-00656449, version 1


Yohann de Castro. Constructions déterministes pour la régression parcimonieuse. Statistiques [math.ST]. Université Paul Sabatier - Toulouse III, 2011. Français. ⟨tel-00656449⟩



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