The document proposes a novel Weakly-supervised Deep Matrix Factorization (WDMF) algorithm for social image tag refinement, assignment and retrieval. WDMF uncovers latent image and tag representations in a latent subspace by exploiting weakly supervised tagging information, visual structure and semantic structure. It can handle noisy, incomplete or subjective tags and noisy or redundant visual features. An optimization problem with a well-defined objective function is formulated and solved using gradient descent with curvilinear search. Extensive experiments on two real-world social image databases demonstrate the effectiveness of the approach.