Regularization techniques augment the objective in optimization formulations for inverse problems with penalty functions -- specified based on prior domain-specific expertise -- to induce a desired structure in the solution. We consider the problem of learning regularizers from data in settings in which precise domain knowledge is not directly available. Previous work under the title of 'dictionary learning' provides regularizers that can be computed via linear programming. We describe a generalization to learn regularizers that can be computed via semidefinite programming. Our approach is based on computing structured factorizations of data matrices that combine recent techniques for rank minimization problems along with the Operator Sinkhorn iteration. (Joint with Yong Sheng Soh)