Sharp Oracle Inequalities for Stationary Points of Nonconvex Penalized M-Estimators

Sharp Oracle Inequalities for Aggregation of Affine Estimators

We consider the problem of combining a (possibly uncountably infinite) set of affine estimators in non-parametric regression model with heteroscedastic Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a PAC-Bayesian type inequality that leads to sharp oracle inequalities in discrete but also in continuous settings. The framework is general enough to cover the combinati...

Sharp Oracle Inequalities for Square Root Regularization

We study a set of regularization methods for high-dimensional linear regression models. These penalized estimators have the square root of the residual sum of squared errors as loss function, and any weakly decomposable norm as penalty function. This fit measure is chosen because of its property that the estimator does not depend on the unknown standard deviation of the noise. On the other hand...

Sharp Oracle Inequalities in Low Rank Estimation

The paper deals with the problem of penalized empirical risk minimization over a convex set of linear functionals on the space of Hermitian matrices with convex loss and nuclear norm penalty. Such penalization is often used in low rank matrix recovery in the cases when the target function can be well approximated by a linear functional generated by a Hermitian matrix of relatively small rank (c...

Asymptotic oracle properties of SCAD-penalized least squares estimators

We study the asymptotic properties of the SCAD-penalized least squares estimator in sparse, high-dimensional, linear regression models when the number of covariates may increase with the sample size. We are particularly interested in the use of this estimator for simultaneous variable selection and estimation. We show that under appropriate conditions, the SCAD-penalized least squares estimator...

Aggregation by Exponential Weighting and Sharp Oracle Inequalities