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.

翻译：过滤的条形码及其代表环编码了数据分析中常用的丰富信息，但其获取过程计算成本高昂。因此，若相关过滤发生微小变化时，开发能更新这些信息的计算方法颇具价值。目前已存在高效算法，可在单纯形交换进入顺序或被添加时更新条形码，但尚未涉及单纯形被移除的情况。我们提出了一种实现方法，当过滤中的单纯形被移除时，能更新缩减边界矩阵。我们的算法——单纯形移除更新过程（SiRUP），可同步更新代表环，并兼容清理优化技术。通过理论与实验方法证明，该算法的复杂度低于从头重新计算条形码，且执行的矩阵列加法次数达到理论最小值。