Identifying progressive mild cognitive impairment (pMCI) patients and predicting when they will convert to Alzheimer's disease (AD) are important for early medical intervention. Multi-modality and longitudinal data provide a great amount of information for improving diagnosis and prognosis. But these data are often incomplete and noisy. To improve the utility of these data for prediction purposes, we propose an approach to denoise the data, impute missing values, and cluster the data into low-dimensional subspaces for pMCI prediction. We assume that the data reside in a space formed by a union of several low-dimensional subspaces and that similar MCI conditions reside in similar subspaces. Therefore, we first use incomplete low-rank representation (ILRR) and spectral clustering to cluster the data according to their representative low-rank subspaces. At the same time, we denoise the data and impute missing values. Then we utilize a low-rank matrix completion (LRMC) framework to identify pMCI patients and their time of conversion. Evaluations using the ADNI dataset indicate that our method outperforms conventional LRMC method.