In this paper, we aim to predict conversion and time-to-conversion of mild cognitive impairment (MCI) patients using multi-modal neuroimaging data and clinical data, via cross-sectional and longitudinal studies. However, such data are often heterogeneous, high-dimensional, noisy, and incomplete. We thus propose a framework that includes sparse feature selection, low-rank affinity pursuit denoising (LRAD), and low-rank matrix completion (LRMC) in this study. Specifically, we first use sparse linear regressions to remove unrelated features. Then, considering the heterogeneity of the MCI data, which can be assumed as a union of multiple subspaces, we propose to use a low rank subspace method (i.e., LRAD) to denoise the data. Finally, we employ LRMC algorithm with three data fitting terms and one inequality constraint for joint conversion and time-to-conversion predictions. Our framework aims to answer a very important but yet rarely explored question in AD study, i.e., when will the MCI convert to AD? This is different from survival analysis, which provides the probabilities of conversion at different time points that are mainly used for global analysis, while our time-to-conversion prediction is for each individual subject. Evaluations using the ADNI dataset indicate that our method outperforms conventional LRMC and other state-of-the-art methods. Our method achieves a maximal pMCI classification accuracy of 84% and time prediction correlation of 0.665.