Mapping the Energy Landscape of Non-convex Optimization Problems

An energy landscape map (ELM) characterizes and visualizes an energy function with a tree structure, in which each leaf node represents a local minimum and each non-leaf node represents the barrier between adjacent energy basins. We demonstrate the utility of ELMs in analyzing non-convex energy minimization problems with two case studies: clustering with Gaussian mixture models and learning mixtures of Bernoulli templates from images. By plotting the ELMs, we are able to visualize the impact of different problem settings on the energy landscape as well as to examine and compare the behaviors of different learning algorithms on the ELMs.