Generalized Concomitant Multi-Task Lasso for sparse multimodal regression
This work addresses a specific issue in high-dimensional regression for multimodal datasets like brain imaging, offering an incremental improvement over existing Concomitant Lasso methods by handling more complex noise structures.
The paper tackles the problem of sparse multimodal regression with heteroscedastic noise, where standard Lasso estimators require known noise levels that are often unknown and vary across data sources. It proposes a jointly convex formulation to estimate regression coefficients and noise covariance, demonstrating improved prediction and support identification in numerical experiments, including applications to M/EEG neuroimaging data.
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, yet the latter is generally unknown in practice. A possible remedy is to consider estimators, such as the Concomitant/Scaled Lasso, which jointly optimize over the regression coefficients as well as over the noise level, making the choice of the regularization independent of the noise level. However, when data from different sources are pooled to increase sample size, or when dealing with multimodal datasets, noise levels typically differ and new dedicated estimators are needed. In this work we provide new statistical and computational solutions to deal with such heteroscedastic regression models, with an emphasis on functional brain imaging with combined magneto- and electroencephalographic (M/EEG) signals. Adopting the formulation of Concomitant Lasso-type estimators, we propose a jointly convex formulation to estimate both the regression coefficients and the (square root of the) noise covariance. When our framework is instantiated to de-correlated noise, it leads to an efficient algorithm whose computational cost is not higher than for the Lasso and Concomitant Lasso, while addressing more complex noise structures. Numerical experiments demonstrate that our estimator yields improved prediction and support identification while correctly estimating the noise (square root) covariance. Results on multimodal neuroimaging problems with M/EEG data are also reported.