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In recent years, machine learning approaches have been successfully applied to the field of neuroimaging for classification and regression tasks. However, many approaches do not give an intuitive relation between the raw features and the diagnosis. Therefore, they are difficult for clinicians to interpret. Moreover, most approaches treat the features extracted from the brain (for example, voxelwise gray matter concentration maps from brain MRI) as independent variables and ignore their spatial and anatomical relations. In this paper, we present a new Support Vector Machine (SVM)-based learning method for the classification of Alzheimer's disease (AD), which integrates spatial-anatomical information. In this way, spatial-neighbor features in the same anatomical region are encouraged to have similar weights in the SVM model. Secondly, we introduce a group lasso penalty to induce structure sparsity, which may help clinicians to assess the key regions involved in the disease. For solving this learning problem, we use an accelerated proximal gradient descent approach. We tested our method on the subset of ADNI data selected by Cuingnet et al. (2011) for Alzheimer's disease classification, as well as on an independent larger dataset from ADNI. Good classification performance is obtained for distinguishing cognitive normals (CN) vs. AD, as well as on distinguishing between various sub-types (e.g. CN vs. Mild Cognitive Impairment). The model trained on Cuignet's dataset for AD vs. CN classification was directly used without re-training to the independent larger dataset. Good performance was achieved, demonstrating the generalizability of the proposed methods. For all experiments, the classification results are comparable or better than the state-of-the-art, while the weight map more clearly indicates the key regions related to Alzheimer's disease.