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The use of machine-learning in neuroimaging offers new perspectives in earlydiagnosis and prognosis of brain diseases. Although such multivariate methodscan capture complex relationships in the data, traditional approaches provideirregular (l2 penalty) or scattered (l1 penalty) predictive pattern with a verylimited relevance. A penalty like Total Variation (TV) that exploits thenatural 3D structure of the images can increase the spatial coherence of theweight map. However, TV penalization leads to non-smooth optimization problemsthat are hard to minimize. We propose an optimization framework that minimizesany combination of l1, l2, and TV penalties while preserving the exact l1penalty. This algorithm uses Nesterov's smoothing technique to approximate theTV penalty with a smooth function such that the loss and the penalties areminimized with an exact accelerated proximal gradient algorithm. We propose anoriginal continuation algorithm that uses successively smaller values of thesmoothing parameter to reach a prescribed precision while achieving the bestpossible convergence rate. This algorithm can be used with other losses orpenalties. The algorithm is applied on a classification problem on the ADNIdataset. We observe that the TV penalty does not necessarily improve theprediction but provides a major breakthrough in terms of support recovery ofthe predictive brain regions.

Predictive support recovery with TV-Elastic Net penalty and logistic regression: an application to structural MRI