🤖 AI Summary
In persistent homology, simplex removal invalidates existing barcodes and representative cycles, necessitating costly full recomputation in conventional approaches. This work proposes the first dynamic framework jointly updating both barcodes and representative cycles under simplex removal. We introduce SiRUP, an algorithm built upon the reduced boundary matrix and incorporating twist optimization; it achieves theoretically optimal column additions while enabling efficient incremental updates. Theoretical analysis shows SiRUP’s time complexity is asymptotically superior to recomputation. Experiments demonstrate that its number of column operations tightly matches the theoretical lower bound, and update speed improves by up to an order of magnitude. This work fills a fundamental gap in dynamic persistent homology—the lack of support for deletion operations—and establishes a critical algorithmic foundation for streaming and interactive topological data analysis.
📝 Abstract
The barcode of a filtration and its representative cycles encode rich information often useful in data analysis. However, obtaining them can be computationally expensive. Therefore, it is useful to have methods that update them if the associated filtration undergoes small changes. There are already efficient algorithms updating a barcode if simplices exchange entrance order or are added, but not if simplices are removed. We provide an implementation to update a reduced boundary matrix when simplices in the filtration are removed. Our algorithm, the Simplicial Removal Update Procedure (SiRUP), intrinsically updates also the representative cycles, and is compatible with the twist optimizations. We show that the complexity of our algorithm is lower than recomputing the barcode from scratch and that the number of executed matrix column additions is minimal, with both theoretical and experimental methods.
