  Marginal optima are minima or maxima of a function with many nearly flat
directions. In settings with many competing optima, marginal ones tend to
attract algorithms and physical dynamics. Often, the important family of
marginal attractors are a vanishing minority compared with nonmarginal optima
and other unstable stationary points. We introduce a generic technique for
conditioning the statistics of stationary points in random landscapes on their
marginality, and apply it in three isotropic settings with qualitatively
different structure: in the spherical spin-glasses, where the energy is
Gaussian and its Hessian is GOE; in multispherical spin glasses, which are
Gaussian but non-GOE; and in sums of squared spherical random functions, which
are non-Gaussian. In these problems we are able to fully characterize the
distribution of marginal optima in the landscape, including when they are in
the minority.
