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The pervasive usage and reach of social media have attracted a surge of attention in the multimedia research community. Community discovery from social media has therefore become an important yet challenging issue. However, due to the subjective generating process, the explicitly observed communities (e.g., group-user and user-user relationship) are often noisy and incomplete in nature. This paper presents a novel approach to discovering communities from social media, including the group membership and user friend structure, by exploring a low-rank matrix recovery technique. In particular, we take Flickr as one exemplary social media platform. We first model the observed indicator matrix of the Flickr community as a summation of a low-rank true matrix and a sparse error matrix. We then formulate an optimization problem by regularizing the true matrix to coincide with the available rich context and content (i.e., photos and their associated tags). An iterative algorithm is developed to recover the true community indicator matrix. The proposed approach leads to a variety of social applications, including community visualization, interest group refinement, friend suggestion, and influential user identification. The evaluations on a large-scale testbed, consisting of 4,919 Flickr users, 1,467 interest groups, and over five million photos, show that our approach opens a new yet effective perspective to solve social network problems with sparse learning technique. Despite being focused on Flickr, our technique can be applied in any other social media community. [ABSTRACT FROM AUTHOR]