Mixup Barcodes: Quantifying Geometric-Topological Interactions between
  Point Clouds
Hubert Wagner Nickolas Arustamyan Matthew Wheeler Peter Bubenik
We combine standard persistent homology with image persistent homology to
define a novel way of characterizing shapes and interactions between them. In
particular, we introduce: (1) a mixup barcode, which captures
geometric-topological interactions (mixup) between two point sets in arbitrary
dimension; (2) simple summary statistics, total mixup and total percentage
mixup, which quantify the complexity of the interactions as a single number;
(3) a software tool for playing with the above.
  As a proof of concept, we apply this tool to a problem arising from machine
learning. In particular, we study the disentanglement in embeddings of
different classes. The results suggest that topological mixup is a useful
method for characterizing interactions for low and high-dimensional data.
Compared to the typical usage of persistent homology, the new tool is sensitive
to the geometric locations of the topological features, which is often
desirable.