  Topological data analysis (TDA) is an area of data science that focuses on
using invariants from algebraic topology to provide multiscale shape
descriptors for geometric data sets such as point clouds. One of the most
important such descriptors is {\em persistent homology}, which encodes the
change in shape as a filtration parameter changes; a typical parameter is the
feature scale. For many data sets, it is useful to simultaneously vary multiple
filtration parameters, for example feature scale and density. While the
theoretical properties of single parameter persistent homology are well
understood, less is known about the multiparameter case. In particular, a
central question is the problem of representing multiparameter persistent
homology by elements of a vector space for integration with standard machine
learning algorithms. Existing approaches to this problem either ignore most of
the multiparameter information to reduce to the one-parameter case or are
heuristic and potentially unstable in the face of noise. In this article, we
introduce a new general representation framework that leverages recent results
on {\em decompositions} of multiparameter persistent homology. This framework
is rich in information, fast to compute, and encompasses previous approaches.
Moreover, we establish theoretical stability guarantees under this framework as
well as efficient algorithms for practical computation, making this framework
an applicable and versatile tool for analyzing geometric and point cloud data.
We validate our stability results and algorithms with numerical experiments
that demonstrate statistical convergence, prediction accuracy, and fast running
times on several real data sets.
