Pruning vineyards: updating barcodes and representative cycles by removing simplices

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 clearing 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.