Simplification of complexes for persistent homology computations (Q2444577)

Computing homology is a heavy job; computing \textit{persistent} homology is still heavier, because it implies computing homology at various levels of a filtration. A convenient method of simplification of a cell complex, called \textit{coreduction}, was introduced years ago for homology [\textit{M. Mrozek} and \textit{B. Batko}, Discrete Comput. Geom. 41, No. 1, 96--118 (2009; Zbl 1163.68041)] as a variation of elementary collapse and adapted to persistence [\textit{M. Mrozek} and \textit{T. Wanner}, Comput. Math. Appl. 60, No. 10, 2812--2833 (2010; Zbl 1207.57001)]. However, the computational burden appears to be still too high. This paper introduces a coreduction technique for computation of persistent homology at positive dimensions, under a fairly natural constraint. A pair \((A, b)\) of cells is a \textit{coreduction pair} if \(b\) is the unique cell in the boundary of \(A\) (possibly after previous coreductions). The coreduction itself consists in the elimination of the pair \((A, b)\) from the cell complex which is the object of the computation. Persistence describes how homology evolves, following a filtration of a given topological space; in most cases, the space is a cell complex and the filtration is the one of sublevel sets of a real map \(g\) defined on it. The constraint considered in this article is that \(g(b) = g(A)\) for each coreduction pair. The coreduction is then proved not to affect persistent homology in positive dimensions (0-persistence can be dealt with separately). A special section is dedicated to the case when \(g(b)\) and \(g(A)\) differ slightly, which is a very common problem in real applications. The paper is very clearly written; particularly well conceived are the simple, ingenuous examples and counterexamples.