Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds
<|reference_start|>Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds: 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.<|reference_end|>
@article{wagner2024mixup,
    title={Mixup Barcodes: Quantifying Geometric-Topological Interactions between
  Point Clouds},
    author={Hubert Wagner, Nickolas Arustamyan, Matthew Wheeler, Peter Bubenik},
    journal={arXiv preprint arXiv:2402.15058},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.15058},
    primaryClass={math.AT cs.CG cs.LG}
}