  In many statistical learning problems, the target functions to be optimized
are highly non-convex in various model spaces and thus are difficult to
analyze. In this paper, we compute \emph{Energy Landscape Maps} (ELMs) which
characterize and visualize 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. The ELM also associates each node with
the estimated probability mass and volume for the corresponding energy basin.
We construct ELMs by adopting the generalized Wang-Landau algorithm and
multi-domain sampler that simulates a Markov chain traversing the model space
by dynamically reweighting the energy function. We construct ELMs in the model
space for two classic statistical learning problems: i) clustering with
Gaussian mixture models or Bernoulli templates; and ii) bi-clustering. We
propose a way to measure the difficulties (or complexity) of these learning
problems and study how various conditions affect the landscape complexity, such
as separability of the clusters, the number of examples, and the level of
supervision; and we also visualize the behaviors of different algorithms, such
as K-mean, EM, two-step EM and Swendsen-Wang cuts, in the energy landscapes.
