Symmetric Tensor Nuclear Norms

This paper studies symmetric tensor nuclear norms. As 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 order and ground field. Lasserre relaxations are proposed for the computation. Theoretical properties of the relaxations are studied. For symmetric tensors, we can compute their nuclear norms, as well as the symmetric nuclear decompositions. The methods can be extended to nonsymmetric tensors.