  The machinery of topological data analysis becomes increasingly popular in a
broad range of machine learning tasks, ranging from anomaly detection and
manifold learning to graph classification. Persistent homology is one of the
key approaches here, allowing us to systematically assess the evolution of
various hidden patterns in the data as we vary a scale parameter. The extracted
patterns, or homological features, along with information on how long such
features persist throughout the considered filtration of a scale parameter,
convey a critical insight into salient data characteristics and data
organization.
  In this work, we introduce two new and easily interpretable topological
summaries for single and multi-parameter persistence, namely, saw functions and
multi-persistence grid functions, respectively. Compared to the existing
topological summaries which tend to assess the numbers of topological features
and/or their lifespans at a given filtration step, our proposed saw and
multi-persistence grid functions allow us to explicitly account for essential
complementary information such as the numbers of births and deaths at each
filtration step.
  These new topological summaries can be regarded as the complexity measures of
the evolving subspaces determined by the filtration and are of particular
utility for applications of persistent homology on graphs. We derive
theoretical guarantees on the stability of the new saw and multi-persistence
grid functions and illustrate their applicability for graph classification
tasks.
