Symmetry & Critical Points for Symmetric Tensor Decomposition Problems
This addresses challenges in tensor decomposition for researchers in optimization and applied mathematics, though it appears incremental as it builds on known symmetry structures.
The paper tackled the nonconvex optimization problem of decomposing symmetric tensors into rank-one terms by leveraging symmetry to construct infinite families of critical points and analyze their properties, revealing complex arrays of saddles and minima with the Hessian index increasing with objective value.
We consider the nonconvex 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 construct infinite families of critical points represented by Puiseux series in the problem dimension, and so obtain precise analytic estimates on the objective function value and the Hessian spectrum. The results enable an analytic characterization of various obstructions to local optimization methods, revealing, in particular, a complex array of saddles and minima that differ in their symmetry, structure, and analytic properties. A notable phenomenon, observed for all critical points considered, concerns the index of the Hessian increasing with the objective function value.