Mohamed Hebiri

On the Prediction Performance of the Lasso

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.

Some theoretical results on the Grouped Variables Lasso

We consider the linear regression model with Gaussian error. We estimate the unknown parameters by a procedure inspired by the Group Lasso estimator introduced in [22]. We show that this estimator satisfies a sparsity inequality, i.e., a bound in terms of the number of non-zero components of the oracle regression vector. We prove that this bound is better, in some cases, than the one achieved by the Lasso and the Dantzig selector.

How Correlations Influence Lasso Prediction

We study how correlations in the design matrix influence Lasso prediction. First, we argue that the higher the correlations, the smaller the optimal tuning parameter. 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.

Rank penalized estimation of a quantum system

We introduce a new method to reconstruct the quantum matrix

Sparse conformal predictors

\bar{\rho}

Confidence Sets for Classification

of a system of

Learning Heteroscedastic Models by Convex Programming under Group Sparsity

n

Generalization of ℓ1 constraints for high dimensional regression problems

-qubits and estimate its rank

Transductive versions of the LASSO and the Dantzig Selector

d

Consistency of plug-in confidence sets for classification in semi-supervised learning

from data obtained by quantum state tomography measurements repeated