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.

翻译：本文研究求解对称张量秩分解关联的高度非凸优化问题。我们构造了损失函数的不变性性质，并证明标准梯度方法检测到的临界点相对于目标张量存在对称性破缺现象。对于不同目标张量与范数选择所呈现的该现象，使得近期发展的解析与代数工具得以应用于研究具有类似对称性破缺特征的非凸优化景观。