Related papers: Steinhaus Filtration and Stable Paths in the Mappe…
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…
The Rips filtration over a finite metric space and its corresponding persistent homology are prominent methods in TDA to summarise the shape of data. Crucial to their use is the bottleneck stability result. A generalisation of the Rips…
Directed graphs arise in many applications where computing persistent homology helps to encode the shape and structure of the input information. However, there are only a few ways to turn the directed graph information into an undirected…
One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…
Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to…
We study the problem of linear feature selection when features are highly correlated. Such settings pose two fundamental challenges. First, how should model similarity be defined? Simply counting features in common can be misleading: two…
Recently, studies on machine learning have focused on methods that use symmetry implicit in a specific manifold as an inductive bias. Grassmann manifolds provide the ability to handle fundamental shapes represented as shape spaces, enabling…
A stable filter has the property that it asymptotically `forgets' initial perturbations. As a result of this property, it is possible to construct approximations of such filters whose errors remain small in time, in other words…
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…
Collaborative filtering (CF) is a popular technique in today's recommender systems, and matrix approximation-based CF methods have achieved great success in both rating prediction and top-N recommendation tasks. However, real-world…
Predictive safety filters enable the integration of potentially unsafe learning-based control approaches and humans into safety-critical systems. In addition to simple constraint satisfaction, many control problems involve additional…
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…
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 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…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
The Persistent Homology Transform (PHT) was introduced in the field of Topological Data Analysis about 10 years ago, and has since been proven to be a very powerful descriptor of Euclidean shapes. The PHT consists of scanning a shape from…
This paper examines the stability of the \`a trous algorithm under arbitrary iteration in the context of a more general study of shift-invariant filter banks. The main results describe sufficient conditions on the associated filters under…
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…
The Persistent Homology Transform (PHT) summarizes a shape in $\mathbb{R}^m$ by collecting persistence diagrams obtained from linear height filtrations in all directions on $\mathbb{S}^{m-1}$. It enjoys strong theoretical guarantees,…
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.…