Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds
This provides a novel method for characterizing interactions between shapes, which could benefit researchers in topological data analysis and machine learning, though it appears incremental as it builds on existing persistent homology techniques.
The authors tackled the problem of quantifying geometric-topological interactions between point clouds by introducing mixup barcodes and summary statistics, and applied this to study disentanglement in embeddings of different classes, suggesting it is useful for low and high-dimensional data.
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.