A remark on the Lasso and the Dantzig selector. (English) Zbl 1489.62210

Summary: This article investigates a new parameter for the high-dimensional regression with noise: the distortion. This latter has attracted a lot of attention recently with the appearance of new deterministic constructions of “almost”-Euclidean sections of the L1-ball. It measures how far is the intersection between the kernel of the design matrix and the unit L1-ball from an L2-ball. We show that the distortion holds enough information to derive oracle inequalities (i.e. a comparison to an ideal situation where one knows the s largest coefficients of the target) for the lasso and the Dantzig selector.


62J07 Ridge regression; shrinkage estimators (Lasso)
62G05 Nonparametric estimation
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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