The Dantzig selector and sparsity oracle inequalities

By Vladimir Koltchinskii. 2009

129 citations recorded by the source corpus.

λ as an estimator of λ ∗ . They called this estimator “the Dantzig selector”. We study the properties of

Subfields: Applied Mathematics, Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/b7c60be760bf9587abeb3896b8e1c46a773fc591

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Citations in the corpus

Some sharp performance bounds for least squares regression with L1 regularization (2009, methodology): “In addition to the above work, prediction error bounds were also obtained in [13, 14] and [11] for general loss functions and random design.”
Statistical significance in high-dimensional linear models (2012, background): “ taking the minimum on the right-handside over all p 1 vectors , we replace it by a constrained minimum, typically over a cone. This leads to the concept of restricted eigenvalues (Bickel et al.,2009;Koltchinskii, 2009a,b;Raskutti et al.,2010) or weaker forms such as the compatibility constants (van”
High-dimensional additive hazards models and the Lasso (2011, background): “For the proof of oracle inequalities with a fast logM/n rate, the restricted eigenvalue (RE) condition introduced in [5] and [15, 16] is of importance.”
The Partial Linear Model in High Dimensions (2013, methodology): “Koltchinskii (2011) and Bühlmann and van de Geer (2011) and references therein) and gives remarkable results in sparse contexts. In this paper we study the high-dimensional partial linear model. Our main contribution is to combine the methods used in Mammen and van de Geer (1997) for the low dimensi”
SIMULTANEOUS ANALYSIS OF LASSO AND DANTZIG SELECTOR (2008, unknown): “Sparsity oracle inequalities for the Dantzig selector with random design are obtained in [15].”
Exponential Screening and optimal rates of sparse estimation (2010, background): “…procedures in terms of the ℓ 0 norm (number of non-zero components) of the estimated vectors or their approximations (see Bickel et al., 2009; Bunea et al., 2007a,b; Candes and Tao, 2007; Koltchinskii, 2008, 2009a,b; van de Geer, 2008; Zhang and Huang, 2008; Zhang, 2009, and references therein).”
Factor models and variable selection in high-dimensional regression analysis (2011, background): “Some important references are Tibshirani (1996), Meinshausen and Bühlmann (2006), Zhao and Yu (2006), van de Geer (2008), Bickel, Ritov and Tsybakov (2009), Candes and Tao (2007) and Koltchinskii (2009).”
A General Theory of Concave Regularization for High-Dimensional Sparse Estimation Problems (2011, methodology): “Refinements of RIP/UUP and SRC were introduced in the literature, such as the restricted eigenvalue of [4, 22],”
Nuclear norm penalization and optimal rates for noisy low rank matrix completion (2010, unknown): “In these problems, they can be expressed in terms of ”restricted eigenvalues“ of certain Gram matrices, cf. the Restricted Eigenvalue condition in [6] for the ﬁxed design case and similar distribution dependent conditions in [16] for the random design case.”
Rate Minimaxity of the Lasso and Dantzig Selector for the lq Loss in lr Balls (2010, background): “The quantity (25) includes as special cases theℓ2 versionRE2,p(ξ,J) of Bickel, Ritov, and Tsybakov (2009) and Koltchinskii (2009) and the constant factorRE1,0(3,J) for the compatibility condition of van de Geer (2007). van de Geer and Bühlmann (2009) calledRE1,0(ξ,J) the restrictedℓ1eigenvalue.”
The adaptive and the thresholded Lasso for potentially misspecified models (and a lower bound for the Lasso) (2011, background): “For a given S, our restricted eigenvalue condition is stronger than the one in [4] or [18, 19].”
On the conditions used to prove oracle results for the Lasso (2009, background): “Such oracle results have been derived by, among others, [4, 19, 21, 2, 16], and for the related Dantzig selector by [11] and [13].”
ℓ1-penalization for mixture regression models (2010, background): “In fact, for the Lasso or Dantzig selector in linear models, restricted eigenvalue conditions (Koltchinskii, 2009; Bickel et al., 2009) are considerably weaker than coherence conditions (Bunea et al.”
High-dimensional covariance matrix estimation with missing observations (2012, background): “For instance, in compressed sensing, it is possible to recover s-sparse vectors of dimension p with only n = O(s log(ep/s)) measurements provided these measurements are carried out properly (see [4,9,12,20,22,24] and the references cited therein for more details).”