Related papers: Kan Approximations of the Persistent Homology Tran…

Persistent homology (PH) is a relatively new field in applied mathematics that studies the components and shapes of discrete data. In this work, we demonstrate that PH can be used as a universal framework to identify phases in spin models,…

3-D shape is important to chemistry, but how important? Machine learning works best when the inputs are simple and match the problem well. Chemistry datasets tend to be very small compared to those generally used in machine learning so we…

In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold, and that they are stable under…

Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self-homeomorphisms of a…

In recent years there has been noticeable interest in the study of the "shape of data". Among the many ways a "shape" could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure:…

Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the…

Let h : R $\rightarrow$ R+ be a lower semi-continuous subbadditive and even function such that h(0) = 0 and h($\theta$) $\ge$ $\alpha$|$\theta$| for some $\alpha$ > 0. The h-mass of a k-polyhedral chain P =$\sum$j $\theta$j$\sigma$j in R n…

We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection $L$ which preserves the persistent diagram of a point cloud $\mathbb{X}$ via…

In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across a filtration. In this work, we extend this setting, and propose the use of bipath persistent homology,…

Let $Z^H= \{Z^H(t), t \in \R^N\}$ be a real-valued $N$-parameter harmonizable fractional stable sheet with index $H = (H_1, \ldots, H_N) \in (0, 1)^N$. We establish a random wavelet series expansion for $Z^H$ which is almost surely…

Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting, exploiting the theory of point…

Visualization in the emerging field of topological data analysis has progressed from persistence barcodes and persistence diagrams to display of two-parameter persistent homology. Although persistence barcodes and diagrams have permitted…

This thesis establishes a generalised setting with which to unify the study of finite local complexity (FLC) patterns. The abstract notion of a "pattern" is introduced, which may be seen as an analogue of the space group of isometries…

We show that for a complete complex algebraic variety the pure component of homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce…

We develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions. We relate directed persistent homology to classical persistent homology, prove some stability results,…

In this note one shows that the four persistence diagrams and measures defined in [5] are particular cases of the maps and the measures discussed in [2] and [1]

This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using the mathematical concept of topology. We explain the fundamental idea of persistent homology from scratch using some examples. We…

Interpolating between measures supported by polygonal or polyhedral domains is a problem that has been recently addressed by the semi-discrete optimal transport framework. Within this framework, one of the domains is discretized with a set…

Persistent homology is a way of determining the topological properties of a data set. It is well known that each persistence module admits the structure of a representation of a finite totally ordered set. In previous work, the authors…

In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to $\R^n$. A natural question arises, whether these approaches are equivalent…