Symmetry Breaking in Symmetric Tensor Decomposition
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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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