  Persistence diagrams are common descriptors of the topological structure of
data appearing in various classification and regression tasks. They can be
generalized to Radon measures supported on the birth-death plane and endowed
with an optimal transport distance. Examples of such measures are expectations
of probability distributions on the space of persistence diagrams. In this
paper, we develop methods for approximating continuous functions on the space
of Radon measures supported on the birth-death plane, as well as their
utilization in supervised learning tasks. Indeed, we show that any continuous
function defined on a compact subset of the space of such measures (e.g., a
classifier or regressor) can be approximated arbitrarily well by polynomial
combinations of features computed using a continuous compactly supported
function on the birth-death plane (a template). We provide insights into the
structure of relatively compact subsets of the space of Radon measures, and
test our approximation methodology on various data sets and supervised learning
tasks.
