Related papers: Fast Computation of Zigzag Persistence
Zigzag persistent homology is a powerful generalisation of persistent homology that allows one not only to compute persistence diagrams with less noise and using less memory, but also to use persistence in new fields of application.…
We introduce a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology. From a zigzag filtration of complexes $(K_i)$, we introduce a zigzag Morse…
Duality results connecting persistence modules for absolute and relative homology provides a fundamental understanding into persistence theory. In this paper, we study similar associations in the context of zigzag persistence. Our main…
Computing persistence over changing filtrations give rise to a stack of 2D persistence diagrams where the birth-death points are connected by the so-called `vines'. We consider computing these vines over changing filtrations for zigzag…
It is well known that ordinary persistence on graphs can be computed more efficiently than the general persistence. Recently, it has been shown that zigzag persistence on graphs also exhibits similar behavior. Motivated by these results, we…
Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features…
Zigzag filtrations of simplicial complexes generalize the usual filtrations by allowing simplex deletions in addition to simplex insertions. The barcodes computed from zigzag filtrations encode the evolution of homological features.…
Vines and vineyard connecting a stack of persistence diagrams have been introduced in the non-zigzag setting by Cohen-Steiner et al. We consider computing these vines over changing filtrations for zigzag persistence while incorporating two…
Boissonnat and Pritam introduced an algorithm to reduce a filtration of flag (or clique) complexes, which can in particular speed up the computation of its persistent homology. They used so-called edge collapse to reduce the input flag…
The barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special…
The classical persistence algorithm computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological…
Most algorithms for computing persistent homology do so by tracking cycles that represent homology classes. There are many choices of such cycles, and specific choices have found different uses in applications. Although it is known that…
We introduce a novel class of finite difference approximations, termed zigzag schemes, that employ a hybrid stencil that is neither symmetrical, nor fully one-sided. These zigzag schemes often enjoy more permissive stability constraints and…
0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…
Persistence modules that decompose into interval modules are important in topological data analysis because we can interpret such intervals as the lifetime of topological features in the data. We can classify the settings in which…
We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…
Matrix reduction is the standard procedure for computing the persistent homology of a filtered simplicial complex with $m$ simplices. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams…
We introduce and study A-infinity persistence of a given homology filtration of topological spaces. This is a family, one for each n > 0, of homological invariants which provide information not readily available by the (persistent) Betti…
This paper introduces an efficient algorithm for persistence diagram computation, given an input piecewise linear scalar field $f$ defined on a $d$-dimensional simplicial complex $K$, with $d \leq 3$. Our work revisits the seminal algorithm…