Related papers: Combinatorial Persistent Homology Transform
As the field of Topological Data Analysis continues to show success in theory and in applications, there has been increasing interest in using tools from this field with methods for machine learning. Using persistent homology, specifically…
Let $K$ be a finite simplicial, cubical, delta or CW complex. The persistence map $\mathrm{PH}$ takes a filter $f:K \rightarrow \mathbb{R}$ as input and returns the barcodes $\mathrm{PH}(f)$ of the associated sublevel set persistent…
An element of a group is called \emph{reversible} if it is conjugate to its inverse. While reversibility in the quaternionic M\"{o}bius group $\mathrm{PSL}(2,\mathbb{H})$ has traditionally been studied using geometric and dynamical methods,…
Multidimensional persistence modules do not admit a concise representation analogous to that provided by persistence diagrams for real-valued functions. However, there is no obstruction for multidimensional persistent Betti numbers to admit…
High-accuracy prediction of the physical properties of amorphous materials is challenging in condensed-matter physics. A promising method to achieve this is machine-learning potentials, which is an alternative to computationally demanding…
This paper aims to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of…
Linear fractional transformations on the extended complex plane are classified up to topological conjugacy. Recall that two transformations f and g are called topologically conjugate if there exists a homeomorphism h such that hg=fh.
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…
In this study, we introduce novel methodologies designed to adapt original data in response to the dynamics of persistence diagrams along Wasserstein gradient flows. Our research focuses on the development of algorithms that translate…
The theory of persistence modules on the commutative ladders $CL_n(\tau)$ provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view…
Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certainfiltering function.…
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…
This short note introduces a geometric representation for binary (or ternary) sequences. The proposed representation is linked to multivariate data plotting according to the radar chart. As an illustrative example, the binary Hamming…
In this paper, we study the geometric decomposition of the degree-$0$ Persistent Homology Transform (PHT) as viewed as a persistence diagram bundle. We focus on star-shaped objects as they can be segmented into smaller, simpler regions…
Persistent homology is a branch of computational algebraic topology that studies shapes and extracts features over multiple scales. In this paper, we present an unsupervised approach that uses persistent homology to study divergent behavior…
In many applications concerning the comparison of data expressed by $\mathbb{R}^m$-valued functions defined on a topological space $X$, the invariance with respect to a given group $G$ of self-homeomorphisms of $X$ is required. While…
Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring…
When persistence diagrams are formalized as the Mobius inversion of the birth-death function, they naturally generalize to the multi-parameter setting and enjoy many of the key properties, such as stability, that we expect in applications.…
Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other…
We prove an extension to the simplicial Nerve Lemma which establishes isomorphism of persistent homology groups, in the case where the covering spaces are filtered. While persistent homology is now widely used in topological data analysis,…