Symmetry Breaking in Symmetric Tensor Decomposition

Yossi Arjevani, Joan Bruna, Michael Field, Joe Kileel, Matthew Trager, Francis Williams

In this note, we consider the optimization problem associated with computing the rank decomposition of a symmetric tensor. We show that, in a well-defined sense, minima in this highly nonconvex optimization problem break the symmetry of the target tensor -- but not too much. This phenomenon of symmetry breaking applies to various choices of tensor norms, and makes it possible to study the optimization landscape using a set of recently-developed symmetry-based analytical tools. The fact that the objective function under consideration is a multivariate polynomial allows us to apply symbolic methods from computational algebra to obtain more refined information on the symmetry breaking phenomenon.

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