Symmetry Breaking in Symmetric Tensor Decomposition
This work addresses a fundamental challenge in tensor decomposition for researchers in optimization and applied mathematics, but it appears incremental as it builds on existing symmetry-breaking phenomena.
The paper tackles the nonconvex optimization problem of symmetric tensor decomposition by analyzing invariance properties and showing that gradient-based methods lead to symmetry-breaking critical points, enabling the application of analytic and algebraic tools to study such landscapes.
In this note, we consider the highly nonconvex optimization problem associated with computing the rank decomposition of symmetric tensors. We formulate the invariance properties of the loss function and show that critical points detected by standard gradient based methods are \emph{symmetry breaking} with respect to the target tensor. The phenomena, seen for different choices of target tensors and norms, make possible the use of recently developed analytic and algebraic tools for studying nonconvex optimization landscapes exhibiting symmetry breaking phenomena of similar nature.