Categorification of persistent homology

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving distance, which we show generalizes the previously-studied bottleneck distance. To illustrate the utility of this approach, we greatly generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of interleavings of these diagrams, and show that if the target category is abelian, so is this category of interleavings.Comment: 27 pages, v3: minor changes, to appear in Discrete & Computational Geometr

Topics in Persistent Homology: From Morse Theory for Minimal Surfaces to Efficient Computation of Image Persistence

We study some problems and develop some theory related to persistent homology, separated into two lines of investigation. In the first part, we introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects in the category of barcodes and the category of pointwise finite-dimensional persistence modules. They also naturally appear in duality results for absolute and relative versions of persistent (co)homology, generalizing previous results in terms of barcodes by de Silva, Morozov, and Vejdemo-Johansson. Due to their functoriality, we can apply these results to morphisms in persistent homology that are induced by morphisms between filtrations. This lays the groundwork for an efficient algorithm to compute barcodes of images and induced matchings of such morphisms, which performs computations in terms of relative cohomology and then translates to absolute homology via the aforementioned dualities. Our method is based on a previous algorithm by Cohen-Steiner, Edelsbrunner, Harer, and Morozov that did not make use of relative cohomology. Using it is crucial, however, because our algorithm applies the clearing optimization introduced by Chen and Kerber, which works particularly well in the context of relative cohomology. We provide an implementation of our algorithm for inclusions of filtrations of Vietoris–Rips complexes in the framework of the software Ripser by Ulrich Bauer. In the second part, we introduce local connectedness conditions on a broad class of functionals that ensure that the persistent homology of their associated sublevel set filtration is q-tame, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the unstable minimal surface theorem given by Morse and Tompkins in terms of persistent Čech homology in a modern and rigorous framework. Moreover, we show that the interleaving distance between the persistent singular homology and the persistent Čech homology of a filtration consisting of paracompact Hausdorff spaces is 0 if it satisfies a similar local connectedness condition to the one used to ensure q-tameness, generalizing a result by Mardešić for locally connected spaces to the setting of filtrations. In contrast to singular homology, the persistent Čech homology of a compact filtration is always upper semi-continuous, which has structural implications in the q-tame case: using a result by Chazal, Crawley-Boevey, and de Silva concerning radicals of persistence modules, we show that every lower semi-continuous q-tame persistence module can be decomposed as a direct sum of interval modules and that every upper semi-continuous q-tame persistence module can be decomposed as a product of interval modules

Homology and Robustness of Level and Interlevel Sets

Given a function f: \Xspace \to \Rspace on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of $f$. In addition, we quantify the robustness of the homology classes under perturbations of $f$ using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case \Xspace = \Rspace^3 has ramifications in the fields of medical imaging and scientific visualization

Quantifying Transversality by Measuring the Robustness of Intersections

By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbations in this space necessary to kill it, and prove that robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings

The observable structure of persistence modules

In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is definable. However, unlike persistence modules indexed over a totally ordered finite set or the natural numbers, such diagrams do not provide a complete invariant of q-tame modules. The purpose of this paper is to show that the category of persistence modules can be adjusted to overcome this issue. We introduce the observable category of persistence modules: a localization of the usual category, in which the classical properties of q-tame modules still hold but where the persistence diagram is a complete isomorphism invariant and all q-tame modules admit an interval decomposition

Structure of Semi-Continuous Q-Tame Persistence Modules

Using a result by Chazal, Crawley-Boevey and de Silva concerning radicals of persistence modules, we show that every lower semi-continuous q-tame persistence module can be decomposed as a direct sum of interval modules and that every upper semi-continuous q-tame persistence module can be decomposed as a product of interval modules.Comment: 11 page

Structural features of persistent homology and their algorithmic transformations

We re-examine the theory and orthodox methods that underlie the study of persistent homology, particularly in its calculation of homological cycle representatives that are associated to persistence diagrams. A common background to the subject covers several aspects: schemes to process input data (embedding it in a low-dimensional manifold), categorical descriptions of persistence objects, and algorithms by which the barcode summarizing the homology is found. We overview these aspects, focusing on altered simplicial complexes, traditional computation of persistent homology, and the stability theorem for barcodes. By reformulating these notions in the language of category theory, we can speak more plainly on some recurring notions that are relevant to our discussion. This ultimately sets up for vector space filtrations that prove to be suitable tools for codifying the homology of complexes, including the (co)images and (co)kernels arising from morphisms of complexes. The main body of work then presents an alternative approach to persistent homology, based on filtrations of vector spaces. We elaborate with an interesting example whose persistent homology is readily computed as a quotient of appropriate filtrations; in the process, we produce a representative basis of homological cycles, a step that is often overlooked in existing literature. The proposed algorithm is also notable in that it easily handles the calculation of (co)images and (co)kernels for persistent morphisms, supplying us with the same level of detail; while other algorithms do exist for computing the barcodes of these universal objects, such methods are not easily generalizable. Finally, we compute appropriate homological cycles and use a certain algorithmic matching scheme that both implies the usual barcode matching and attempts to better interpret this interesting behavior

Towards Stratification Learning through Homology Inference

A topological approach to stratification learning is developed for point cloud data drawn from a stratified space. Given such data, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space, giving a stratification for each radius level. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into different strata, and we prove a result which guarantees the correctness of our clustering, given certain topological conditions; some geometric intuition for these topological conditions is also provided. Our correctness result is then given a probabilistic flavor: we give bounds on the minimum number of sample points required to infer, with probability, which points belong to the same strata. Finally, we give an explicit algorithm for the clustering, prove its correctness, and apply it to some simulated data.Comment: 48 page