  We show that recent results on randomized dimension reduction schemes that
exploit structural properties of data can be applied in the context of
persistent homology. In the spirit of compressed sensing, the dimension
reduction is determined by the Gaussian width of a structure associated to the
data set, rather than its size, and such a reduction can be computed
efficiently. We further relate the Gaussian width to the doubling dimension of
a finite metric space, which appears in the study of the complexity of other
methods for approximating persistent homology. We can therefore literally
replace the ambient dimension by an intrinsic notion of dimension related to
the structure of the data.
