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Robust Subspace Clustering
, 2013
"... Subspace clustering refers to the task of finding a multisubspace representation that best fits a collection of points taken from a highdimensional space. This paper introduces an algorithm inspired by sparse subspace clustering (SSC) [17] to cluster noisy data, and develops some novel theory demo ..."
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Cited by 22 (1 self)
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Subspace clustering refers to the task of finding a multisubspace representation that best fits a collection of points taken from a highdimensional space. This paper introduces an algorithm inspired by sparse subspace clustering (SSC) [17] to cluster noisy data, and develops some novel theory demonstrating its correctness. In particular, the theory uses ideas from geometric functional analysis to show that the algorithm can accurately recover the underlying subspaces under minimal requirements on their orientation, and on the number of samples per subspace. Synthetic as well as real data experiments complement our theoretical study, illustrating our approach and demonstrating its effectiveness.
Fused sparsity and robust estimation for linear models with unknown variance
 In NIPS
, 2012
"... with unknown variance ..."
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Path Thresholding: Asymptotically TuningFree HighDimensional Sparse Regression
"... In this paper, we address the challenging problem of selecting tuning parameters for highdimensional sparse regression. We propose a simple and computationally efficient method, called path thresholding (PaTh), that transforms any tuning parameterdependent sparse regression algorithm into an ..."
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In this paper, we address the challenging problem of selecting tuning parameters for highdimensional sparse regression. We propose a simple and computationally efficient method, called path thresholding (PaTh), that transforms any tuning parameterdependent sparse regression algorithm into an asymptotically tuningfree sparse regression algorithm. More specifically, we prove that, as the problem size becomes large (in the number of variables and in the number of observations), PaTh performs accurate sparse regression, under appropriate conditions, without specifying a tuning parameter. In finitedimensional settings, we demonstrate that PaTh can alleviate the computational burden of model selection algorithms by significantly reducing the search space of tuning parameters. 1
Estimator selection in the Gaussian setting
 SUBMITTED TO THE ANNALES DE L’INSTITUT HENRI POINCARE ́ PROBABILITÉS ET STATISTIQUES
, 2012
"... We consider the problem of estimating the mean f of a Gaussian vector Y with independent components of common unknown variance σ2. Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection F of estimators of f based on Y a ..."
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We consider the problem of estimating the mean f of a Gaussian vector Y with independent components of common unknown variance σ2. Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection F of estimators of f based on Y and, with the same data Y, aim at selecting an estimator among F with the smallest Euclidean risk. No assumptions on the estimators are made and their dependencies with respect to Y may be unknown. We establish a nonasymptotic risk bound for the selected estimator and derive oracletype inequalities when F consists of linear estimators. As particular cases, our approach allows to handle the problems of aggregation, model selection as well as those of choosing a window and a kernel for estimating a regression function, or tuning the parameter involved in a penalized criterion. In all theses cases but aggregation, the method can be easily implemented. For illustration, we carry out two simulation studies. One aims at comparing our procedure to crossvalidation for choosing a tuning parameter. The other shows how to implement our approach to solve the problem of variable selection in practice.
A Global Homogeneity Test for HighDimensional Linear Regression
, 2013
"... Abstract: This paper is motivated by the comparison of genetic networks based on microarray samples. The aim is to test whether the differences observed between two inferred Gaussian graphical models come from real differences or arise from estimation uncertainties. Adopting a neighborhood approach, ..."
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Abstract: This paper is motivated by the comparison of genetic networks based on microarray samples. The aim is to test whether the differences observed between two inferred Gaussian graphical models come from real differences or arise from estimation uncertainties. Adopting a neighborhood approach, we consider a twosample linear regression model with random design and propose a procedure to test whether these two regressions are the same. Relying on multiple testing and variable selection strategies, we develop a testing procedure that applies to highdimensional settings where the number of covariates p is larger than the number of observations n1 and n2 of the two samples. Both type I and type II errors are explicitely controlled from a nonasymptotic perspective and the test is proved to be minimax adaptive to the sparsity. The performances of the test are evaluated on simulated data. Moreover, we illustrate how this procedure can be used to compare genetic networks on Hess et al breast cancer microarray dataset.
Introduction to the Special Issue on Sparsity and Regularization Methods
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