Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913135205548032 |
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| author | Wagner, Hubert Arustamyan, Nickolas Wheeler, Matthew Bubenik, Peter |
| author_facet | Wagner, Hubert Arustamyan, Nickolas Wheeler, Matthew Bubenik, Peter |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds Wagner, Hubert Arustamyan, Nickolas Wheeler, Matthew Bubenik, Peter Algebraic Topology Computational Geometry Machine Learning 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. |
| title | Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds |
| topic | Algebraic Topology Computational Geometry Machine Learning |
| url | https://arxiv.org/abs/2402.15058 |