  What is the "right" topological invariant of a large point cloud X? Prior
research has focused on estimating the full persistence diagram of X, a
quantity that is very expensive to compute, unstable to outliers, and far from
a sufficient statistic. We therefore propose that the correct invariant is not
the persistence diagram of X, but rather the collection of persistence diagrams
of many small subsets. This invariant, which we call "distributed persistence,"
is perfectly parallelizable, more stable to outliers, and has a rich inverse
theory. The map from the space of point clouds (with the quasi-isometry metric)
to the space of distributed persistence invariants (with the
Hausdorff-Bottleneck distance) is a global quasi-isometry. This is a much
stronger property than simply being injective, as it implies that the inverse
of a small neighborhood is a small neighborhood, and is to our knowledge the
only result of its kind in the TDA literature. Moreover, the quasi-isometry
bounds depend on the size of the subsets taken, so that as the size of these
subsets goes from small to large, the invariant interpolates between a purely
geometric one and a topological one. Lastly, we note that our inverse results
do not actually require considering all subsets of a fixed size (an enormous
collection), but a relatively small collection satisfying certain covering
properties that arise with high probability when randomly sampling subsets.
These theoretical results are complemented by two synthetic experiments
demonstrating the use of distributed persistence in practice.
