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We show that the high-dimensional behavior of symmetrically penalized least squares with a possibly non-separable, symmetric, convex penalty in both (i) the Gaussian sequence model and (ii) the linear model with uncorrelated Gaussian designs nearly matches the behavior of least squares with an appropriately chosen separable penalty in these same models. The similarity in behavior is precisely quantified by a finite-sample concentration inequality in both cases. Our results help clarify the role non-separability can play in high-dimensional M-estimation. In particular, if the empirical distribution of the coordinates of the parameter is known --exactly or approximately-- there are at most limited advantages to using non-separable, symmetric penalties over separable ones. In contrast, if the empirical distribution of the coordinates of the parameter is unknown, we argue that non-separable, symmetric penalties automatically implement an adaptive procedure which we characterize. We also provide a partial converse which characterizes adaptive procedures which can be implemented in this way.
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
We study the problem of exact support recovery based on noisy observations and present Refined Least Squares (RLS). Given a set of noisy measurement $$ myvec{y} = myvec{X}myvec{theta}^* + myvec{omega},$$ and $myvec{X} in mathbb{R}^{N times D}$ which
In model selection, several types of cross-validation are commonly used and many variants have been introduced. While consistency of some of these methods has been proven, their rate of convergence to the oracle is generally still unknown. Until now,
We consider a nonparametric version of the integer-valued GARCH(1,1) model for time series of counts. The link function in the recursion for the variances is not specified by finite-dimensional parameters, but we impose nonparametric smoothness condi
We study the performance of the Least Squares Estimator (LSE) in a general nonparametric regression model, when the errors are independent of the covariates but may only have a $p$-th moment ($pgeq 1$). In such a heavy-tailed regression setting, we s