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Computing topological persistence for simplicial maps
, 2012
"... Algorithms for persistent homology and zigzag persistent homology are wellstudied for homology modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence underZ2 coefficients for a sequence of general simplicial maps. First, ..."
Abstract

Cited by 10 (2 self)
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Algorithms for persistent homology and zigzag persistent homology are wellstudied for homology modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence underZ2 coefficients for a sequence of general simplicial maps. First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provides an algorithm for computing the persistence induced by simplicial maps. Our main result is that the above simple minded approach can be improved for a sequence of simplicial maps given in a monotone direction. The improvement results from the use of the socalled annotations that we show can determine the persistence of simplicial maps using a lighter data structure. A consistent annotation through atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally. Finally, we exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blowup in size.
The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology
"... Persistent homology with coefficients in a field F coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors with the simplices. We separate the representation of the simplic ..."
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Cited by 5 (1 self)
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Persistent homology with coefficients in a field F coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors with the simplices. We separate the representation of the simplicial complex from the representation of the cohomology groups, and introduce a new data structure for maintaining the annotation matrix, which is more compact and reduces substancially the amount of matrix operations. In addition, we propose a heuristic to simplify further the representation of the cohomology groups and improve both time and space complexities. The paper provides a theoretical analysis, as well as a detailed experimental study of our implementation and comparison with stateoftheart software for persistent homology and cohomology.
Homology Annotations via Matrix Reduction
"... In this work, we propose an alternative algorithm for annotating simplices of a simplicial complex K with subbases of a basis B of its pdimensional homology group H1(K). Such annotations, summed over psimplices in any pcycle z, provide an expression of z in B. This allows us to answer queries of ..."
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In this work, we propose an alternative algorithm for annotating simplices of a simplicial complex K with subbases of a basis B of its pdimensional homology group H1(K). Such annotations, summed over psimplices in any pcycle z, provide an expression of z in B. This allows us to answer queries of null homology and independence of pcycles efficiently and improve the running time of the greedy algorithm to computea shortest basis of H1(K). The best known algorithm for the shortest basis problem that does not use annotations has a time complexity of O(n 4), where n is the size of the 2skeleton of K. We improve it to O(n 3 +n 2 g 2), where g is the rank of H1(K). Annotating simplices with a homology basis has been considered before [1]. The existing approachcan preprocess the simplicial complex and assign annotations in subcubictime. However, this involves computing the LSPdecomposition of the boundary matrix, which can be computationally cumbersome. We present a simple and implementationfriendly O(n 3) approach that fits nicely to the family of matrix reduction algorithms such as the persistence algorithm and the classic Smith normal form reduction. Our analysis also reveals interesting connections to the persistence algorithm. Namely, our matrix reduction method computes pairing between simplices under homomorphisms between homology groups that are not necessarily induced by inclusions between the subcomplexes of the filtration of K. 1.