  Big data often has emergent structure that exists at multiple levels of
abstraction, which are useful for characterizing complex interactions and
dynamics of the observations. Here, we consider multiple levels of abstraction
via a multiresolution geometry of data points at different granularities. To
construct this geometry we define a time-inhomogeneous diffusion process that
effectively condenses data points together to uncover nested groupings at
larger and larger granularities. This inhomogeneous process creates a deep
cascade of intrinsic low pass filters on the data affinity graph that are
applied in sequence to gradually eliminate local variability while adjusting
the learned data geometry to increasingly coarser resolutions. We provide
visualizations to exhibit our method as a continuously-hierarchical clustering
with directions of eliminated variation highlighted at each step. The utility
of our algorithm is demonstrated via neuronal data condensation, where the
constructed multiresolution data geometry uncovers the organization, grouping,
and connectivity between neurons.
