Related papers: Filtration Simplification for Persistent Homology …
Persistent homology is a tool that can be employed to summarize the shape of data by quantifying homological features. When the data is an object in $\mathbb{R}^d$, the (augmented) persistent homology transform ((A)PHT) is a family of…
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 enables fast and computable comparison of topological objects. However, it is naturally limited to the analysis of topological spaces. We extend the theory of persistence, by guaranteeing robustness and computability to…
The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth…
Persistent Homology is a widely used topological data analysis tool that creates a concise description of the topological properties of a point cloud based on a specified filtration. Most filtrations used for persistent homology depend…
We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…
High order networks are weighted hypergraphs col- lecting relationships between elements of tuples, not necessarily pairs. Valid metric distances between high order networks have been defined but they are difficult to compute when the…
Single-parameter persistent homology, a key tool in topological data analysis, has been widely applied to data problems along with statistical techniques that quantify the significance of the results. In contrast, statistical techniques for…
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…
The machinery of topological data analysis becomes increasingly popular in a broad range of machine learning tasks, ranging from anomaly detection and manifold learning to graph classification. Persistent homology is one of the key…
The persistent homology with coefficients in a field F coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors…
We propose a study of multipartite entanglement through persistent homology, a tool used in topological data analysis. In persistent homology, a 1-parameter filtration of simplicial complexes called persistence complex is used to reveal…
In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph…
Persistent Homology is a powerful tool in Topological Data Analysis (TDA) to capture topological properties of data succinctly at different spatial resolutions. For graphical data, shape, and structure of the neighborhood of individual data…
Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly…
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,…
Determining whether two graphs are isomorphic is a fundamental problem with practical applications in areas such as molecular chemistry or social network analysis, yet it remains a challenging task, with exact solutions often being…
Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have…
Persistent homology, an algebraic method for discerning structure in abstract data, relies on the construction of a sequence of nested topological spaces known as a filtration. Two-parameter persistent homology allows the analysis of data…
Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying…