Incomplete dataset due to missing values is ubiquitous in multimodal neuroimaging data. Denoting an incomplete dataset as a feature matrix, where each row contains feature values of the multi-modality data of a sample, we propose a framework to predict the corresponding interrelated multiple target outputs (e.g., diagnosis label and clinical scores) from this feature matrix. This is achieved by applying a matrix completion algorithm on a shrunk version of the feature matrix that is augmented with the corresponding target output matrix, to simultaneously predict the missing feature values and the unknown target outputs. We shrink the matrix by first partition the large incomplete feature matrix into smaller submatrices that contain complete feature data. Treating each target output prediction from the submatrix as a task, we perform multi-task learning based feature and sample selections to select the most discriminative features and samples from each submatrix. Features and samples which are not selected from any of the submatrices are removed, resulting in a shrunk feature matrix, which is still incomplete. This shrunk matrix together with its corresponding target matrix (of possibly unknown values) are finally simultaneously completed using a low rank matrix completion algorithm. Experimental results using the ADNI dataset indicate that our proposed framework yields better identification accuracy at higher speed compared with conventional imputation-based identification methods.