Barcodes of Towers and a Streaming Algorithm for Persistent Homology

A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger than the tower and can be efficiently computed by a streaming algorithm, both in theory and in practice. Furthermore, we show that our approach can be combined with a streaming algorithm to compute the barcode of the tower via matrix reduction. The space complexity of the algorithm does not depend on the length of the tower, but the maximal size of any subcomplex within the tower. Experimental evaluations show that our approach can efficiently handle towers with billions of complexes

Improved Approximate Rips Filtrations with Shifted Integer Lattices

Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes constitutes an expensive task because of a combinatorial explosion in the complex size. For n points in R^d, we present a scheme to construct a 4.24-approximation of the multi-scale filtration of the Rips complex in the L-infinity metric, which extends to a O(d^{0.25})-approximation of the Rips filtration for the Euclidean case. The k-skeleton of the resulting approximation has a total size of n2^{O(d log k)}. The scheme is based on the integer lattice and on the barycentric subdivision of the d-cube

Computing Persistent Homology of Flag Complexes via Strong Collapses

In this article, we focus on the problem of computing Persistent Homology of a flag tower, i.e. a sequence of flag complexes connected by simplicial maps. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. We show that strong collapses of flag complexes can be computed in time O(k^2v^2) where v is the number of vertices of the complex and k is the maximal degree of its graph. Moreover we can strong collapse a flag complex knowing only its 1-skeleton and the resulting complex is also a flag complex. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the core flag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremely efficient

Strong Collapse for Persistence

We introduce a fast and memory efficient approach to compute the persistent homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by J. Barmak and E. Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one. Our approach has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for other types of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot of space and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel. As a result and as demonstrated by numerous experiments on publicly available data sets, our approach is extremely fast and memory efficient in practice

Edge Collapse and Persistence of Flag Complexes

In this article, we extend the notions of dominated vertex and strong collapse of a simplicial complex as introduced by J. Barmak and E. Miniam. We say that a simplex (of any dimension) is dominated if its link is a simplicial cone. Domination of edges appears to be a very powerful concept, especially when applied to flag complexes. We show that edge collapse (removal of dominated edges) in a flag complex can be performed using only the 1-skeleton of the complex. Furthermore, the residual complex is a flag complex as well. Next we show that, similar to the case of strong collapses, we can use edge collapses to reduce a flag filtration ? to a smaller flag filtration ?^c with the same persistence. Here again, we only use the 1-skeletons of the complexes. The resulting method to compute ?^c is simple and extremely efficient and, when used as a preprocessing for persistence computation, leads to gains of several orders of magnitude w.r.t the state-of-the-art methods (including our previous approach using strong collapse). The method is exact, irrespective of dimension, and improves performance of persistence computation even in low dimensions. This is demonstrated by numerous experiments on publicly available data

Computing Zigzag Vineyard Efficiently Including Expansions and Contractions

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 more operations: expansions and contractions in addition to the transpositions considered in the non-zigzag setting. Although expansions and contractions can be implemented in quadratic time in the non-zigzag case by utilizing the linear-time transpositions, it is not obvious how they can be carried out under the zigzag framework with the same complexity. While transpositions alone can be easily conducted in linear time using the recent FastZigzag algorithm, expansions and contractions pose difficulty in breaking the barrier of cubic complexity. Our main result is that, the half-way constructed up-down filtration in the FastZigzag algorithm indeed can be used to achieve linear time complexity for transpositions and quadratic time complexity for expansions and contractions, matching the time complexity of all corresponding operations in the non-zigzag case