Handling correlated and repeated measurements with the smoothed multivariate square-root Lasso
This work addresses a specific limitation in Lasso-type estimators for researchers in fields like neuroimaging, but it is incremental as it builds on existing concomitant methods to handle more complex noise scenarios.
The paper tackles the problem of selecting optimal regularization parameters in high-dimensional regression with correlated and repeated measurements by proposing a concomitant estimator that handles complex noise structures without averaging data. The method demonstrates practical benefits on toy datasets, simulated data, and real neuroimaging data, though no specific numerical results are provided.
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.