Topological persistence has proven to be a key concept for the study of real-valued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case of continuous functions defined over triangulable spaces.In this paper, we present new stability results that do not suffer from the above restrictions. Furthermore, by working at an algebraic level directly, we make it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence. Along the way, we extend the definition of persistence diagram to a larger setting, introduce the notions of discretization of a persistence module and associated pixelization map, define a proximity measure between persistence modules, and show how to interpolate between persistence modules, thereby lending a more analytic character to this otherwise algebraic setting. We believe these new theoretical concepts and tools shed new light on the theory of persistence, in addition to simplifying proofs and enabling new applications.

We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a new notation for calculations on quiver representations. We show that the stringent finiteness conditions required by traditional methods are not necessary to prove the existence and stability of the persistence diagram. We introduce weaker hypotheses for taming persistence modules, which are met in practice and are strong enough for the theory still to work. The constructions and proofs enabled by our framework are, we claim, cleaner and simpler.‚ We introduce 'decorated' real numbers to remove ambiguity about interval endpoints.Decorations are also what make the measure theory work.‚ We define several kinds of 'tameness' for a persistence module. These occur naturally in practice. The most restrictive of these, finite type, is what is normally seen in the literature. We show how to work effectively with the less restrictive hypotheses.‚ We introduce a special notation for calculations on quiver representations. This considerably simplifies the linear algebra (for instance, in proving the 'box lemma').Our goal in introducing these ideas is to enable other authors to define persistence diagrams cleanly, in a wide variety of situations, without imposing unnecessary restrictions (such as assuming a function to be Morse). For instance, it will be seen in forthcoming work that the approach here can be used to define the levelset zigzag persistence of [5] quite broadly.This paper owes much to [7] (and its published journal version [6]), which established the existence and stability of persistence diagrams for modules whose persistence maps are of finite rank. In the present work we call these modules 'q-tame'.Traditionally, continuous persistence diagrams have been treated in one of two ways. Most commonly, one makes the aggressive assumption that the situation being studied has only finitely many 'critical values'. Alternatively, as carried out in [7] for q-tame modules, the diagram is constructed using a careful limiting process through ever-finer discretisations of the parameter. The former strategy may be appropriate when working with real-world data, where every persistence module really is finite in every way, but it excludes very common theoretical situations. The latter strategy was devised to overcome these restrictions, but unfortunately the limiting arguments turn out to be quite complicated. Our new approach gives the best of both worlds; we are able to work with broader classes of persistence modules, and we can reason about their diagrams in a clean way using arguments of a finite nature.The other debt to [7] is the recasting of stability as a statement about interleaved persistence modules, the so-called 'algebraic stability theorem'. In this paper we re-recast the result as a statement about 1-parameter families of measures. This allows us to prove stability ...

We present a clustering scheme that combines a mode-seeking phase with a cluster merging phase in the corresponding density map. While mode detection is done by a standard graph-based hill-climbing scheme, the novelty of our approach resides in its use of topological persistence to guide the merging of clusters. Our algorithm provides additional feedback in the form of a set of points in the plane, called a persistence diagram (PD), which provably reflects the prominences of the modes of the density. In practice, this feedback enables the user to choose relevant parameter values, so that under mild sampling conditions the algorithm will output the correct number of clusters, a notion that can be made formally sound within persistence theory. In addition, the output clusters have the property that their spatial locations are bound to the ones of the basins of attraction of the peaks of the density. The algorithm only requires rough estimates of the density at the data points, and knowledge of (approximate) pairwise distances between them. It is therefore applicable in any metric space. Meanwhile, its complexity remains practical: although the size of the input distance matrix may be up to quadratic in the number of data points, a careful implementation only uses a linear amount of memory and takes barely more time to run than to read through the input.