Benign cubic and quartic forms on the sphere

Abstract

One considers the NP-hard problem of minimizing real cubic and quartic forms on the Euclidean unit sphere. One first extends the Hessian spectral characterization of local minima by the author. In particular one provides a gap-depth characterization of negative global and spurious minima, and proves that the minimum value of an odd degree-d form is recovered (up to the factor d(d -1)) by minimizing the smallest eigenvalue of its ambient Hessian over the sphere (and equivalently the unit ball). Next, one exhibits several classes of benign families of cubic and quartic forms (i.e. with no spurious local minima on the sphere) and prove that some are robust to small coefficient perturbation (i.e. the resulting form remains benign even if it leaves the initial family). Therefore the class of benign cubic and quartic forms is full-dimensional (in the ambient space of coefficients), for n $\ge$ 2. Also for every odd degree d, the class of benign degree-d forms increases with the dimension (however it does not imply that such forms become more frequent when the dimension increases).