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arXiv · 2402.15058

Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds

Hubert Wagner·Nickolas Arustamyan·Matthew Wheeler·Peter Bubenik
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Abstract

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.

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Hubert Wagner, Nickolas Arustamyan, Matthew Wheeler, Peter Bubenik. 2024-02-23. Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds. https://arxiv.org/abs/2402.15058

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