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

The use of machine-learning in neuroimaging offers new perspectives in early diagnosis and prognosis of brain diseases. Although such multivariate methods can capture complex relationships in the data, traditional approaches provide irregular (ℓ
2
penalty) or scattered (£1 penalty) predictive pattern with a very limited relevance. A penalty like Total Variation (TV) that exploits the natural 3D structure of the images can increase the spatial coherence of the weight map. However, TV penalization leads to non-smooth optimization problems that are hard to minimize. We propose an optimization framework that minimizes any combination of ℓ
1
, ℓ
2
, and TV penalties while preserving the exact ℓ
1
penalty. This algorithm uses Nesterov's smoothing technique to approximate the TV penalty with a smooth function such that the loss and the penalties are minimized with an exact accelerated proximal gradient algorithm. We propose an original continuation algorithm that uses successively smaller values of the smoothing parameter to reach a prescribed precision while achieving the best possible convergence rate. This algorithm can be used with other losses or penalties. The algorithm is applied on a classification problem on the ADNI dataset. We observe that the TV penalty does not necessarily improve the prediction but provides a major breakthrough in terms of support recovery of the predictive brain regions.