The $k$-support norm is a regularizer which has been successfully applied to sparse vector prediction problems. We show that it belongs to a general class of norms which can be formulated as a parameterized infimum over quadratics. We further extend
the $k$-support norm to matrices, and we observe that it is a special case of the matrix cluster norm. Using this formulation we derive an efficient algorithm to compute the proximity operator of both norms. This improves upon the standard algorithm for the $k$-support norm and allows us to apply proximal gradient methods to the cluster norm. We also describe how to solve regularization problems which employ center
