Parallelism, uniqueness, and large-sample asymptotics for the Dantzig selector.
نویسندگان
چکیده
The Dantzig selector (Candès and Tao, 2007) is a popular ℓ1-regularization method for variable selection and estimation in linear regression. We present a very weak geometric condition on the observed predictors which is related to parallelism and, when satisfied, ensures the uniqueness of Dantzig selector estimators. The condition holds with probability 1, if the predictors are drawn from a continuous distribution. We discuss the necessity of this condition for uniqueness and also provide a closely related condition which ensures uniqueness of lasso estimators (Tibshirani, 1996). Large sample asymptotics for the Dantzig selector, i.e. almost sure convergence and the asymptotic distribution, follow directly from our uniqueness results and a continuity argument. The limiting distribution of the Dantzig selector is generally non-normal. Though our asymptotic results require that the number of predictors is fixed (similar to (Knight and Fu, 2000)), our uniqueness results are valid for an arbitrary number of predictors and observations.
منابع مشابه
Multi-Stage Dantzig Selector
We consider the following sparse signal recovery (or feature selection) problem: given a design matrix X ∈ Rn×m (m À n) and a noisy observation vector y ∈ R satisfying y = Xβ∗ + 2 where 2 is the noise vector following a Gaussian distribution N(0, σI), how to recover the signal (or parameter vector) β∗ when the signal is sparse? The Dantzig selector has been proposed for sparse signal recovery w...
متن کاملDASSO: Connections Between the Dantzig Selector and Lasso
We propose a new algorithm, DASSO, for fitting the entire coefficient path of the Dantzig selector with a similar computational cost to the LARS algorithm that is used to compute the Lasso. DASSO efficiently constructs a piecewise linear path through a sequential simplex-like algorithm, which is remarkably similar to LARS. Comparison of the two algorithms sheds new light on the question of how ...
متن کاملThe Constrained Dantzig Selector with Enhanced Consistency
The Dantzig selector has received popularity for many applications such as compressed sensing and sparse modeling, thanks to its computational efficiency as a linear programming problem and its nice sampling properties. Existing results show that it can recover sparse signals mimicking the accuracy of the ideal procedure, up to a logarithmic factor of the dimensionality. Such a factor has been ...
متن کاملThe Dantzig Selector in Cox’s Proportional Hazards Model
The Dantzig Selector is a recent approach to estimation in high-dimensional linear regression models with a large number of explanatory variables and a relatively small number of observations. As in the least absolute shrinkage and selection operator (LASSO), this approach sets certain regression coefficients exactly to zero, thus performing variable selection. However, such a framework, contra...
متن کاملDantzig selector homotopy with dynamic measurements
The Dantzig selector is a near ideal estimator for recovery of sparse signals from linear measurements in the presence of noise. It is a convex optimization problem which can be recast into a linear program (LP) for real data, and solved using some LP solver. In this paper we present an alternative approach to solve the Dantzig selector which we call “Primal Dual pursuit” or “PD pursuit”. It is...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- The Canadian journal of statistics = Revue canadienne de statistique
دوره 41 1 شماره
صفحات -
تاریخ انتشار 2013