Concomitant Lasso with Repetitions (CLaR): beyond averaging multiple realizations of heteroscedastic noise

Quentin Bertrand (1) , Mathurin Massias (1) , Alexandre Gramfort (1) , Joseph Salmon (2)

1 PARIETAL - Modelling brain structure, function and variability based on high-field MRI data
2 IMAG - Institut Montpelliérain Alexander Grothendieck

Sparsity promoting norms are frequently used in high dimensional regression. A limitation of such Lasso-type estimators is that the optimal regularization parameter depends on the unknown noise level. Estimators such as the concomitant Lasso address this dependence by jointly estimating the noise level and the regression coefficients. Additionally, in many applications, the data is obtained by averaging multiple measurements: this reduces the noise variance, but it dramatically reduces sample sizes and prevents refined noise modeling. In this work, we propose a concomitant estimator that can cope with complex noise structure by using non-averaged measurements. The resulting optimization problem is convex and amenable, thanks to smoothing theory, to state-of-the-art optimization techniques that leverage the sparsity of the solutions. Practical benefits are demonstrated on toy datasets, realistic simulated data and real neuroimaging data.

Domaines: Machine Learning [stat.ML]; Optimisation et contrôle [math.OC]; Applications [stat.AP]

Quentin Bertrand, Mathurin Massias, Alexandre Gramfort, Joseph Salmon. Concomitant Lasso with Repetitions (CLaR): beyond averaging multiple realizations of heteroscedastic noise. 2019. ⟨hal-02010014v2⟩