Related papers: Computing Persistent Homology with Various Coeffic…
Persistent homology computes the multiscale topology of a data set by using a sequence of discrete complexes. In this paper, we propose that persistent homology may be a useful tool for studying the structure of the landscape of string…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
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…
To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital…
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…
The aim of this paper is to present a method for computation of persistent homology that performs well at large filtration values. To this end we introduce the concept of filtered covers. We show that the persistent homology of a bounded…
In this paper, for a finite group, we discuss a method for calculating equivariant homology with constant coefficients. We apply it to completely calculate the geometric fixed points of the equivariant spectrum representing equivariant…
Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram --…
We present in this paper a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces,…
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…
Persistent homology analysis, a recently developed computational method in algebraic topology, is applied to the study of the phase transitions undergone by the so-called XY-mean field model and by the phi^4 lattice model, respectively. For…
Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques.…
Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that utilize persistence diagrams in statistical inference, a full Bayesian treatment remains…
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…
Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of…
Computing Persistent Homology for large point clouds remains a bottleneck for the wider adoption of persistent homology by the scientific community. We present an algorithm which can compute the degree-1 Vietoris-Rips Persistent Homology of…
In this article we propose a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes,…
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 extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input…
Matrix reduction is the standard procedure for computing the persistent homology of a filtered simplicial complex with $m$ simplices. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams…