Recently, there have been great interests for computer-aided diagnosis of Alzheimer's disease (AD) and its prodromal stage, mild cognitive impairment (MCI). Unlike the previous methods that considered simple low-level features such as gray matter tissue volumes from MRI, and mean signal intensities from PET, in this paper, we propose a deep learning-based latent feature representation with a stacked auto-encoder (SAE). We believe that there exist latent non-linear complicated patterns inherent in the low-level features such as relations among features. Combining the latent information with the original features helps build a robust model in AD/MCI classification, with high diagnostic accuracy. Furthermore, thanks to the unsupervised characteristic of the pre-training in deep learning, we can benefit from the target-unrelated samples to initialize parameters of SAE, thus finding optimal parameters in fine-tuning with the target-related samples, and further enhancing the classification performances across four binary classification problems: AD vs. healthy normal control (HC), MCI vs. HC, AD vs. MCI, and MCI converter (MCI-C) vs. MCI non-converter (MCI-NC). In our experiments on ADNI dataset, we validated the effectiveness of the proposed method, showing the accuracies of 98.8, 90.7, 83.7, and 83.3 % for AD/HC, MCI/HC, AD/MCI, and MCI-C/MCI-NC classification, respectively. We believe that deep learning can shed new light on the neuroimaging data analysis, and our work presented the applicability of this method to brain disease diagnosis.