  Diffusion condensation is a dynamic process that yields a sequence of
multiscale data representations that aim to encode meaningful abstractions. It
has proven effective for manifold learning, denoising, clustering, and
visualization of high-dimensional data. Diffusion condensation is constructed
as a time-inhomogeneous process where each step first computes and then applies
a diffusion operator to the data. We theoretically analyze the convergence and
evolution of this process from geometric, spectral, and topological
perspectives. From a geometric perspective, we obtain convergence bounds based
on the smallest transition probability and the radius of the data, whereas from
a spectral perspective, our bounds are based on the eigenspectrum of the
diffusion kernel. Our spectral results are of particular interest since most of
the literature on data diffusion is focused on homogeneous processes. From a
topological perspective, we show diffusion condensation generalizes
centroid-based hierarchical clustering. We use this perspective to obtain a
bound based on the number of data points, independent of their location. To
understand the evolution of the data geometry beyond convergence, we use
topological data analysis. We show that the condensation process itself defines
an intrinsic condensation homology. We use this intrinsic topology as well as
the ambient persistent homology of the condensation process to study how the
data changes over diffusion time. We demonstrate both types of topological
information in well-understood toy examples. Our work gives theoretical
insights into the convergence of diffusion condensation, and shows that it
provides a link between topological and geometric data analysis.
