The fusion of complementary information contained in multi-modality data [e.g., magnetic resonance imaging (MRI), positron emission tomography (PET), and genetic data] has advanced the progress of automated Alzheimer's disease (AD) diagnosis. However, multi-modality based AD diagnostic models are often hindered by the missing data, i.e., not all the subjects have complete multi-modality data. One simple solution used by many previous studies is to discard samples with missing modalities. However, this significantly reduces the number of training samples, thus leading to a sub-optimal classification model. Furthermore, when building the classification model, most existing methods simply concatenate features from different modalities into a single feature vector without considering their underlying associations. As features from different modalities are often closely related (e.g., MRI and PET features are extracted from the same brain region), utilizing their inter-modality associations may improve the robustness of the diagnostic model. To this end, we propose a novel latent representation learning method for multi-modality based AD diagnosis. Specifically, we use all the available samples (including samples with incomplete modality data) to learn a latent representation space. Within this space, we not only use samples with complete multi-modality data to learn a common latent representation, but also use samples with incomplete multi-modality data to learn independent modality-specific latent representations. We then project the latent representations to the label space for AD diagnosis. We perform experiments using 737 subjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database, and the experimental results verify the effectiveness of our proposed method.