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Topological Data Analysis (TDA) is a rising field of computational topology in which the topological structure of a data set can be observed by persistent homology. By considering a sequence of sublevel sets, one obtains a filtration that…
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine…
Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their $0$-dimensional homology. While this area has been substantially studied, we present a new approach to…
Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…
Persistent Homology (PH) is a fundamental tool in computational topology, designed to uncover the intrinsic geometric and topological features of data across multiple scales. Originating within the broader framework of Topological Data…
Weighted digraphs are used to model a variety of natural systems and can exhibit interesting structure across a range of scales. In order to understand and compare these systems, we require stable, interpretable, multiscale descriptors. To…
We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then…
Persistent homology captures the evolution of topological features of a model as a parameter changes. The most commonly used summary statistics of persistent homology are the barcode and the persistence diagram. Another summary statistic,…
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…
Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the \v{C}ech or Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers.…
We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale…
Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and…
Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex $K$. These topological changes are summarized in persistence diagrams. We propose…
Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single…
Finding an optimal parameter of a black-box function is important for searching stable material structures and finding optimal neural network structures, and Bayesian optimization algorithms are widely used for the purpose. However, most of…
In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction…
Persistent homology studies the evolution of k-dimensional holes along a nested sequence of simplicial complexes (called a filtration). The set of bars (i.e. intervals) representing birth and death times of k-dimensional holes along such…
This paper presents a new clustering algorithm for space-time data based on the concepts of topological data analysis and in particular, persistent homology. Employing persistent homology - a flexible mathematical tool from algebraic…
The machine learning technique of persistent homology classifies complex systems or datasets by computing their topological features over a range of characteristic scales. There is growing interest in applying persistent homology to…
Topological data analysis uses tools from topology -- the mathematical area that studies shapes -- to create representations of data. In particular, in persistent homology, one studies one-parameter families of spaces associated with data,…