  We present a novel framework exploiting the cascade of phase transitions
occurring during a simulated annealing of the Expectation-Maximisation
algorithm to cluster datasets with multi-scale structures. Using the weighted
local covariance, we can extract, a posteriori and without any prior knowledge,
information on the number of clusters at different scales together with their
size. We also study the linear stability of the iterative scheme to derive the
threshold at which the first transition occurs and show how to approximate the
next ones. Finally, we combine simulated annealing together with recent
developments of regularised Gaussian mixture models to learn a principal graph
from spatially structured datasets that can also exhibit many scales.
