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.