  This paper considers the problem of understanding the exit time for
trajectories of gradient-related first-order methods from saddle neighborhoods
under some initial boundary conditions. Given the 'flat' geometry around saddle
points, first-order methods can struggle to escape these regions in a fast
manner due to the small magnitudes of gradients encountered. In particular,
while it is known that gradient-related first-order methods escape
strict-saddle neighborhoods, existing analytic techniques do not explicitly
leverage the local geometry around saddle points in order to control behavior
of gradient trajectories. It is in this context that this paper puts forth a
rigorous geometric analysis of the gradient-descent method around strict-saddle
neighborhoods using matrix perturbation theory. In doing so, it provides a key
result that can be used to generate an approximate gradient trajectory for any
given initial conditions. In addition, the analysis leads to a linear exit-time
solution for gradient-descent method under certain necessary initial
conditions, which explicitly bring out the dependence on problem dimension,
conditioning of the saddle neighborhood, and more, for a class of strict-saddle
functions.
