The vast user-provided image tags on the popular photo sharing websites may greatly facilitate image retrieval and management. However, these tags are often imprecise and/or incomplete, resulting in unsatisfactory performances in tag related applications. In this work, the tag refinement problem is formulated as a decomposition of the user-provided tag matrix D into a low-rank refined matrix A and a sparse error matrix E, namely D = A + E, targeting the optimality measured by four aspects: 1) low-rank: A is of low-rank owing to the semantic correlations among the tags; 2) content consistency: if two images are visually similar, their tag vectors (i.e., column vectors of A) should also be similar; 3) tag correlation: if two tags co-occur with high frequency in general images, their co-occurrence frequency (described by two row vectors of A) should also be high; and 4) error sparsity: the matrix E is sparse since the tag matrix D is sparse and also humans can provide reasonably accurate...