With the increasing prevalence of Alzheimer's disease, research focuses on the early computer-aided diagnosis of dementia with the goal to understand the disease process, determine risk and preserving factors, and explore preventive therapies. By now, large amounts of data from multi-site studies have been made available for developing, training, and evaluating automated classifiers. Yet, their translation to the clinic remains challenging, in part due to their limited generalizability across different datasets. In this work, we describe a compact classification approach that mitigates overfitting by regularizing the multinomial regression with the mixed ℓ1/ℓ2 norm. We combine volume, thickness, and anatomical shape features from MRI scans to characterize neuroanatomy for the three-class classification of Alzheimer's disease, mild cognitive impairment and healthy controls. We demonstrate high classification accuracy via independent evaluation within the scope of the CADDementia challenge. We, furthermore, demonstrate that variations between source and target datasets can substantially influence classification accuracy. The main contribution of this work addresses this problem by proposing an approach for supervised domain adaptation based on instance weighting. Integration of this method into our classifier allows us to assess different strategies for domain adaptation. Our results demonstrate (i) that training on only the target training set yields better results than the naïve combination (union) of source and target training sets, and (ii) that domain adaptation with instance weighting yields the best classification results, especially if only a small training component of the target dataset is available. These insights imply that successful deployment of systems for computer-aided diagnostics to the clinic depends not only on accurate classifiers that avoid overfitting, but also on a dedicated domain adaptation strategy.