Generalized Concomitant Multi-Task Lasso for Sparse Multimodal Regression

In high dimension, it is customary to consider Lasso-type estimators to enforce sparsity. For standard Lasso theory to hold, the regularization parameter should be proportional to the noise level, which is often unknown in practice. A remedy is to consider estimators such as the Concomitant Lasso, which jointly optimize over the regression coefficients and the noise level. However, when data from different sources are pooled to increase sample size, noise levels differ and new dedicated estimators are needed. We provide new statistical and computational solutions to perform heteroscedastic regression, with an emphasis on brain imaging with magnetoand electroencephalography (M/EEG). When instantiated to de-correlated noise, our framework leads to an efficient algorithm whose computational cost is not higher than for the Lasso, but addresses more complex noise structures. Experiments demonstrate improved prediction and support identification with correct estimation of noise levels.