  We introduce graphcodes, a novel multi-scale summary of the topological
properties of a dataset that is based on the well-established theory of
persistent homology. Graphcodes handle datasets that are filtered along two
real-valued scale parameters. Such multi-parameter topological summaries are
usually based on complicated theoretical foundations and difficult to compute;
in contrast, graphcodes yield an informative and interpretable summary and can
be computed as efficient as one-parameter summaries. Moreover, a graphcode is
simply an embedded graph and can therefore be readily integrated in machine
learning pipelines using graph neural networks. We describe such a pipeline and
demonstrate that graphcodes achieve better classification accuracy than
state-of-the-art approaches on various datasets.
