Fast Computation of Zigzag Persistence

Abstract

Zigzag persistence is a powerful extension of the standard persistence which allows deletions of simplices besides insertions. However, computing zigzag persistence usually takes considerably more time than the standard persistence. This paper sheds light on narrowing this efficiency gap. Our main result is that an input simplex-wise zigzag filtration can be converted to a cell -wise non-zigzag filtration of a ∆-complex with the same length, where the cells are copies of the input simplices. This conversion step incurs very little cost. Furthermore, the barcode of the original filtration can be easily read from the barcode of the new cell-wise filtration because the conversion embodies a series of diamond switches known in topological data analysis. This seemingly simple observation opens up the vast possibilities for improving the computation of zigzag persistence because any efficient algorithm/software for standard persistence can now be applied to computing zigzag persistence. Our experiment shows that this indeed achieves substantial performance gain over the existing state-of-the-art softwares.

Temporal network analysis using zigzag persistence

“…Until recently, there has been limited involvement of zigzag persistence in these types of applications since even though in theory the running time should be similar to standard persistence [48], in practice it has not seen the flurry of optimizations available in the regular case [49,50]. However, recent work [51][52][53] promises substantial improvements in the potentially available code, which should further make the tools discussed in this paper more accessible to a wide array of data sets.…”