Symmetric Tensor Nuclear Norms

This paper studies nuclear norms of symmetric tensors. As recently shown by Friedland and Lim, the nuclear norm of a symmetric tensor can be achieved at a symmetric decomposition. We discuss how to compute symmetric tensor nuclear norms, depending on the tensor order and the ground field. Lasserre relaxations are proposed for the computation. The theoretical properties of the relaxations are studied. For symmetric tensors, we can compute their nuclear norms, as well as the nuclear decompositions. The proposed methods can be extended to nonsymmetric tensors.