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：我们考虑与实对称张量分解为秩一张量之和相关的非凸优化问题。利用丰富的对称结构，推导出临界点族的Puiseux级数表示，从而获得临界值及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面临的各种几何障碍，特别揭示出由对称性、结构及解析性质差异导致的复杂鞍点与局部极小点阵列。所有被考虑的临界点中出现的理想现象是：点的指标（即Hessian负特征值数目）随目标函数值增加而增加。最后，利用牛顿多面体论证，给出固定对称性下所有临界点的完整枚举，并表明：与在不同张量范数选择下保持不变的全局极小点集不同，某些非全局极小点族会涌现，而另一些则会消失。