Abstract. This paper introduces a generalized forward-backward splitting algorithm for finding a zero of a sum of maximal monotone operators B + n i=1 A i , where B is cocoercive. It involves the computation of B in an explicit (forward) step and of the parallel computation of the resolvents of the A i 's in a subsequent implicit (backward) step. We prove its convergence in infinite dimension, and robustness to summable errors on the computed operators in the explicit and implicit steps. In particular, this allows efficient minimization of the sum of convex functions f + n i=1 g i , where f has a Lipschitz-continuous gradient and each g i is simple in the sense that its proximity operator is easy to compute. The resulting method makes use of the regularity of f in the forward step, and the proximity operators of the g i 's are applied in parallel in the backward step. While the forwardbackward algorithm cannot deal with more than n = 1 non-smooth function, we generalize it to the case of arbitrary n. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.

In this paper, we introduce a mathematical framework for obtaining spatially smooth semantic labelings of 3D point clouds from a pointwise classification. We argue that structured regularization offers a more versatile alternative to the standard graphical model approach. Indeed, our framework allows us to choose between a wide range of fidelity functions and regularizers, influencing the properties of the solution. In particular, we investigate the conditions under which the smoothed labeling remains probabilistic in nature, allowing us to measure the uncertainty associated with each label. Finally, we present efficient algorithms to solve the corresponding optimization problems.To demonstrate the performance of our approach, we present classification results derived for standard benchmark datasets. We demonstrate that the structured regularization framework offers higher accuracy at a lighter computational cost in comparison to the classic graphical model approach.

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In the context of Alzheimer's disease, two challenging issues are (1) the characterization of local hippocampal shape changes specific to disease progression and (2) the identification of mild-cognitive impairment patients likely to convert. In the literature, (1) is usually solved first to detect areas potentially related to the disease. These areas are then considered as an input to solve (2). As an alternative to this sequential strategy, we investigate the use of a classification model using logistic regression to address both issues (1) and (2) simultaneously. The classification of the patients therefore does not require any a priori definition of the most representative hippocampal areas potentially related to the disease, as they are automatically detected. We first quantify deformations of patients' hippocampi between two time points using the large deformations by diffeomorphisms framework and transport these deformations to a common template. Since the deformations are expected to be spatially structured, we perform classification combining logistic loss and spatial regularization techniques, which have not been explored so far in this context, as far as we know. The main contribution of this paper is the comparison of regularization techniques enforcing the coefficient maps to be spatially smooth (Sobolev), piecewise constant (total variation) or sparse (fused LASSO) with standard regularization techniques which do not take into account the spatial structure (LASSO, ridge and ElasticNet). On a dataset of 103 patients out of ADNI, the techniques using spatial regularizations lead to the best classification rates. They also find coherent areas related to the disease progression.