Persistent Homology and Nested Dissection

Abstract

Nested dissection exploits the underlying topology to do matrix reductions while persistent homology exploits matrix reductions to the reveal underlying topology. It seems natural that one should be able to combine these techniques to beat the currently best bound of matrix multiplication time for computing persistent homology. However, nested dissection works by fixing a reduction order, whereas persistent homology generally constrains the ordering according to an input filtration. Despite this obstruction, we show that it is possible to combine these two theories. This shows that one can improve the computation of persistent homology if the underlying space has some additional structure. We give reasonable geometric conditions under which one can beat the matrix multiplication bound for persistent homology.1 Introduction Persistent homology [ELZ02] has become the standard tool in the growing field of topological data analysis (TDA). The underlying idea is to consider a sequence of increasing shapes and track the homological changes of the shapes in this process, that is, the appearance and disappearance of holes. This information can be read off from a (generalized) LU -decomposition of the boundary matrix which is a combinatorial representation of the sequence of shapes. All persistent homology algorithms used in practice are based on matrix reduction through row or column operations on an initially sparse boundary matrix. Due to matrix fill-in, a cubic complexity in the size of the matrix can be achieved on worst-case examples [Mor05]. On the other hand, algorithms often show near-linear asymptotic behavior on most realistic inputs. It is likely that the structure of realistic examples keeps the matrices sparse during the computation, but how can we quantify this?Generalized nested dissection [LRT79] is a technique to reduce the effect of fill-in in Gaussian elimination and related matrix reduction problems for instances which