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.
Tensor Networks for Latent Variable Analysis. Part I: Algorithms for Tensor Train Decomposition
Brain Science Institute, RIKEN（RIKEN脑科学研究所）；Systems Research Institute PAS（波兰科学院系统研究所）；University of Bonn（波恩大学）；Institute of Information Theory and Automation（信息与自动化研究所）；Georgetown University（乔治城大学）；Imperial College（帝国理工学院）
An Explicit SOS Decomposition of A Fourth Order Four Dimensional Hankel Tensor with A Symmetric Generating Vector
School of Mathematics and Statistics, Zhengzhou University（郑州大学数学与统计学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
