Although the Lasso has been extensively studied, the relationship between its prediction performance and the correlations of the covariates is not fully understood. In this paper, we give new insights into this relationship in the context of multiple linear regression. We show, in particular, that the incorporation of a simple correlation measure into the tuning parameter can lead to a nearly optimal prediction performance of the Lasso even for highly correlated covariates. However, we also reveal that for moderately correlated covariates, the prediction performance of the Lasso can be mediocre irrespective of the choice of the tuning parameter. We finally show that our results also lead to near-optimal rates for the least-squares estimator with total variation penalty.

We consider a linear regression problem in a high dimensional setting where the number of covariates p can be much larger than the sample size n. In such a situation, one often assumes sparsity of the regression vector, i.e., the regression vector contains many zero components. We propose a Lasso-type estimatorβ Quad (where 'Quad' stands for quadratic) which is based on two penalty terms. The first one is the ℓ 1 norm of the regression coefficients used to exploit the sparsity of the regression as done by the Lasso estimator, whereas the second is a quadratic penalty term introduced to capture some additional information on the setting of the problem. We detail two special cases: the Elastic-Netβ EN introduced in [42], which deals with sparse problems where correlations between variables may exist; and the Smooth-Lasso 1βSL , which responds to sparse problems where successive regression coefficients are known to vary slowly (in some situations, this can also be interpreted in terms of correlations between successive variables). From a theoretical point of view, we establish variable selection consistency results and show that β Quad achieves a Sparsity Inequality, i.e., a bound in terms of the number of non-zero components of the 'true' regression vector. These results are provided under a weaker assumption on the Gram matrix than the one used by the Lasso. In some situations this guarantees a significant improvement over the Lasso. Furthermore, a simulation study is conducted and shows that the S-Lassoβ SL performs better than known methods as the Lasso, the Elastic-Netβ EN , and the Fused-Lasso (introduced in [30]) with respect to the estimation accuracy. This is especially the case when the regression vector is 'smooth', i.e., when the variations between successive coefficients of the unknown parameter of the regression are small. The study also reveals that the theoretical calibration of the tuning parameters and the one based on 10 fold cross validation imply two S-Lasso solutions with close performance.

We study how correlations in the design matrix influence Lasso prediction. First, we argue that the higher the correlations are, the smaller the optimal tuning parameter is. This implies in particular that the standard tuning parameters, that do not depend on the design matrix, are not favorable. Furthermore, we argue that Lasso prediction works well for any degree of correlations if suitable tuning parameters are chosen. We study these two subjects theoretically as well as with simulations.

We introduce a new method to reconstruct the density matrix ρ of a system of n-qubits and estimate its rank d from data obtained by quantum state tomography measurements repeated m times. The procedure consists in minimizing the risk of a linear estimatorρ of ρ penalized by given rank (from 1 to 2 n ), whereρ is previously obtained by the moment method. We obtain simultaneously an estimator of the rank and the resulting density matrix associated to this rank. We establish an upper bound for the error of penalized estimator, evaluated with the Frobenius norm, which is of order dn(4/3) n /m and consistency for the estimator of the rank. The proposed methodology is computationaly efficient and is illustrated with some example states and real experimental data sets.

We consider the linear regression model with Gaussian error. We estimate the unknown parameters by a procedure inspired from the Group Lasso estimator introduced in [21]. We show that this estimator satisfies a sparsity oracle inequality, i.e., a bound in terms of the number of non-zero components of the oracle vector. We prove that this bound is better, in some cases, than the one achieved by the Lasso and the Dantzig selector.