Efficient Computation of Image Persistence

Abstract

We present an algorithm for computing the barcode of the image of a morphism in persistent homology induced by an inclusion of filtered finite-dimensional chain complexes. The algorithm makes use of the clearing optimization and can be applied to inclusion-induced maps in persistent absolute homology and persistent relative cohomology for filtrations of pairs of simplicial complexes. The clearing optimization works particularly well in the context of relative cohomology, and using previous duality results we can translate the barcodes of images in relative cohomology to those in absolute homology. This forms the basis for our implementation of image persistence computations for inclusions of filtrations of Vietoris-Rips complexes in the framework of the software Ripser. matching construction yields a partial bijection between the barcodes of H * (L • ) and H * (K • ), which can be used to bound the distance between these two barcodes from above. The induced matching is defined in terms of the image persistence barcode, i.e., the barcode of im H * (f • ), motivating the problem of computing this barcode. A first algorithm for this problem has been given by for the special case where f • is of the formThe authors also propose a method for getting rid of the intersection assumption using a mapping cylinder construction. This construction, however, might not be computationally feasible and we will show that this additional step is in fact not necessary.Cohen-Steiner et al. [12] propose applications of image persistence for recovering the persistent homology of a noisy function on a noisy domain, see also the related work by Chazal et al. [7]. Recently, Reani and Bobrowski [15] proposed a method that includes the computation of induced matchings in order to pair up common topological features in different data sets, with applications to statistical bootstrapping. Furthermore, the computation of image barcodes is used in a distributed algorithm for persistent homology based on the Mayer-Vietoris spectral sequence by Álvaro Torras Casas [6]. Image persistence of endomorphisms such as Steenrod squares on the persistent (co)homology of a single filtration has also been proposed by Lupo et al. [14] as a tool to get more comprehensive invariants than the standard persistent (co)homology barcodes.Despite the usefulness of image persistence, there are a few aspects that have prevented these techniques from being widely used in applications so far. Specifically, to the best of our knowledge, there is no publicly available implementation at this moment. Furthermore, computation using the known algorithms is slow in comparison to modern algorithms for a single filtration. Indeed, computing usual persistent homology for larger data sets arising in real-world applications only became feasible in recent years due to optimizations that exploit various structural properties and algebraic identities of the problem [9, 16, 1]. Our goal is to adapt these speed-ups to the computation of images and induced matchings.The basic algorithm ...

Fast topological signal identification and persistent cohomological cycle matching

“…Both of these persistence diagrams may be computed applying a matrix reduction algorithm to appropriate boundary matrices. Bauer and Schmahl (2022) also show that the clearing algorithm implemented in Ripser (see Sect. 1.4) can be applied to compute image-persistence.…”

“…• We specialize the definition of interval matching proposed by Reani and Bobrowski (2021b) to simplex-wise filtrations (see Definition 15), making it compatible with the output of Ripser-image (Bauer and Schmahl 2022) and more widely applicable.…”

“…Their work is precisely motivated by the extension of the duality results by de Silva et al (2011a) to the setting of image-persistence. A version of a result from Bauer and Schmahl ( 2021) is then revisited in Bauer and Schmahl (2022) in Proposition 3.12 in order to implement Ripser-image, which plays a central role in our work. This result will be further explained in Sect.…”

A Theory of Sub-Barcodes

“…Image persistence has found fruitful use in scalar field analysis [12] and clustering in manifolds [31]. More recently, Bauer and Schmahl developed an efficient algorithm for computing image persistence [32].…”

“…However, when the bounds are sufficiently close in a neighborhood of significant topological features, one expects to see the relevant features represented in the sub-barcode. Importantly, the barcode B G L can be computed using the image persistence algorithm of Cohen-Steiner et al [29] (see also [32]), providing a natural computational variant of Corollary 4.2.…”