  Despite much attention, the comparison of reduced-dimension representations
of high-dimensional data remains a challenging problem in multiple fields,
especially when representations remain high-dimensional compared to sample
size. We offer a framework for evaluating the topological similarity of
high-dimensional representations of very high-dimensional data, a regime where
topological structure is more likely captured in the distribution of
topological "noise" than a few prominent generators. Treating each
representational map as a metric embedding, we compute the Vietoris-Rips
persistence of its image. We then use the topological bootstrap to analyze the
re-sampling stability of each representation, assigning a "prevalence score"
for each nontrivial basis element of its persistence module. Finally, we
compare the persistent homology of representations using a prevalence-weighted
variant of the Wasserstein distance. Notably, our method is able to compare
representations derived from different samples of the same distribution and, in
particular, is not restricted to comparisons of graphs on the same vertex set.
In addition, representations need not lie in the same metric space. We apply
this analysis to a cross-sectional sample of representations of functional
neuroimaging data in a large cohort and hierarchically cluster under the
prevalence-weighted Wasserstein. We find that the ambient dimension of a
representation is a stronger predictor of the number and stability of
topological features than its decomposition rank. Our findings suggest that
important topological information lies in repeatable, low-persistence homology
generators, whose distributions capture important and interpretable differences
between high-dimensional data representations.

Computational Geometry;Image and Video Processing;Algebraic Topology;Neurons and Cognition