  Persistent homology is an important methodology from topological data
analysis which adapts theory from algebraic topology to data settings and has
been successfully implemented in many applications. It produces a statistical
summary in the form of a persistence diagram, which captures the shape and size
of the data. Despite its widespread use, persistent homology is simply
impossible to implement when a dataset is very large. In this paper we address
the problem of finding a representative persistence diagram for prohibitively
large datasets. We adapt the classical statistical method of bootstrapping,
namely, drawing and studying smaller multiple subsamples from the large
dataset. We show that the mean of the persistence diagrams of subsamples --
taken as a mean persistence measure computed from the subsamples -- is a valid
approximation of the true persistent homology of the larger dataset. We give
the rate of convergence of the mean persistence diagram to the true persistence
diagram in terms of the number of subsamples and size of each subsample. Given
the complex algebraic and geometric nature of persistent homology, we adapt the
convexity and stability properties in the space of persistence diagrams
together with random set theory to achieve our theoretical results for the
general setting of point cloud data. We demonstrate our approach on simulated
and real data, including an application of shape clustering on complex
large-scale point cloud data.
