Adjacency Criterion For Gradient Flow With Multiple Local Maxima
For researchers studying gradient flows and simulated annealing, this work offers a theoretical tool to understand the geometry of complex optimization landscapes, though it is incremental in nature.
This paper establishes an adjacency criterion for gradient flows with multiple local maxima, providing a necessary and sufficient condition for two regions of attraction to be adjacent. Applied to a specific flow with n! local maxima, it characterizes equilibria, computes indices, and identifies adjacent pairs, then uses this to model a stochastic double bracket flow.
In this paper, we investigate the geometry of a general class of gradient flows with multiple local maxima. we decompose the underlying space into disjoint regions of attraction and establish the adjacency criterion. The criterion states a necessary and sufficient condition for two regions of attraction of stable equilibria to be adjacent. We then apply this criterion on a specific type of gradient flow which has as many as n! local maxima. In particular, we characterize the set of equilibria, compute the index of each critical manifold and moreover, find all pairs of adjacent neighbors. As an application of the adjacency criterion, we introduce a stochastic version of the double bracket flow and set up a Markov model to approximate the sample path behavior. The study of this specific prototype with its special structure provides insight into many other difficult problems involving simulated annealing.