Non-sufficiency of smoothness in the gradient conjecture

Abstract

It is well known that for analytic cost functions, gradient flow trajectories have finite length and converge to a single critical point. The gradient conjecture of R. Thom states that, again for analytic cost functions, whenever the gradient flow trajectory converges, the limit of its unit secants exists. One might think that already the convergence of the gradient flow trajectory to a critical point is enough to ensure that the unit secants have a limit, but this does not hold in general - the gradient conjecture is to a certain extend sharp. We provide a counterexample in case of the missing analyticity assumption, that is a smooth (non-analytic) cost function $f$, where the limit of unit secants does not exist. In addition, $f$ satisfies even a strong geometric length-distance convergence property.