We consider the non-convex optimization problem associated with the decomposition of a real symmetric tensor into a sum of rank one terms. Use is made of the rich symmetry structure to derive Puiseux series representations of families of critical points, and so obtain precise analytic estimates on the critical values and the Hessian spectrum. The sharp results make possible an analytic characterization of various geometric obstructions to local optimization methods, revealing in particular a complex array of saddles and local minima which differ by their symmetry, structure and analytic properties. A desirable phenomenon, occurring for all critical points considered, concerns the index of a point, i.e., the number of negative Hessian eigenvalues, increasing with the value of the objective function. Lastly, a Newton polytope argument is used to give a complete enumeration of all critical points of fixed symmetry, and it is shown that contrarily to the set of global minima which remains invariant under different choices of tensor norms, certain families of non-global minima emerge, others disappear.

翻译：本文研究实对称张量分解为秩一张量之和相关的非凸优化问题。利用丰富的对称性结构，推导出临界点族的Puiseux级数表示，从而获得关于临界值和Hessian谱的精确解析估计。这些精确结果使得能够解析刻画局部优化方法中的各种几何障碍，尤其揭示了一类复杂的鞍点和局部极小值点，这些点在对称性、结构及解析性质上存在差异。在所有考虑的临界点中，出现了一个值得关注的现象：点的指标（即Hessian负特征值的数量）随目标函数值的增加而增加。最后，采用Newton多面体论证对固定对称性的所有临界点进行完整枚举，并证明：与保持张量范数选择不变的整体极小点集不同，某些非整体极小点族会出现，而另一些则会消失。