  In this paper, we consider topological featurizations of data defined over
simplicial complexes, like images and labeled graphs, obtained by convolving
this data with various filters before computing persistence. Viewing a
convolution filter as a local motif, the persistence diagram of the resulting
convolution describes the way the motif is distributed across the simplicial
complex. This pipeline, which we call convolutional persistence, extends the
capacity of topology to observe patterns in such data. Moreover, we prove that
(generically speaking) for any two labeled complexes one can find some filter
for which they produce different persistence diagrams, so that the collection
of all possible convolutional persistence diagrams is an injective invariant.
This is proven by showing convolutional persistence to be a special case of
another topological invariant, the Persistent Homology Transform. Other
advantages of convolutional persistence are improved stability, greater
flexibility for data-dependent vectorizations, and reduced computational
complexity for certain data types. Additionally, we have a suite of experiments
showing that convolutions greatly improve the predictive power of persistence
on a host of classification tasks, even if one uses random filters and
vectorizes the resulting diagrams by recording only their total persistences.
