相关论文: Linear-Size Approximations to the Vietoris-Rips Fi…
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.…
Topological data analysis, as a tool for extracting topological features and characterizing geometric shapes, has experienced significant development across diverse fields. Its key mathematical techniques include persistent homology and the…
We consider the topological and geometric reconstruction of a geodesic subspace of $\mathbb{R}^N$ both from the \v{C}ech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages…
For a metric space $(X, d)$ and a scale parameter $r \geq 0$, the Vietoris-Rips complex $\mathcal{VR}(X;r)$ is a simplicial complex on vertex set $X$, where a finite set $\sigma \subseteq X$ is a simplex if and only if diameter of $\sigma$…
We extend classical tools from rational homotopy theory to topological data analysis by introducing persistent Sullivan minimal models of persistent topological spaces. Our main result establishes that the interleaving distance between such…
The computation of Vietoris-Rips persistence barcodes is both execution-intensive and memory-intensive. In this paper, we study its computational structure and identify several unique mathematical properties and algorithmic opportunities…
Although transformer-based models have been dominating the field of deep learning, various studies of their embedding space have shown that they suffer from "representation degeneration problem": embeddings tend to be distributed in a…
Detecting the dimension of a hidden manifold from a point sample has become an important problem in the current data-driven era. Indeed, estimating the shape dimension is often the first step in studying the processes or phenomena…
We present a geometric perspective on sparse filtrations used in topological data analysis. This new perspective leads to much simpler proofs, while also being more general, applying equally to Rips filtrations and Cech filtrations for any…
Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. the Cech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from…
Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams…
We consider the degree-Rips construction from topological data analysis, which provides a density-sensitive, multiparameter hierarchical clustering algorithm. We analyze its stability to perturbations of the input data using the…
Persistent homology is constrained to purely topological persistence while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low-dimensional multiscale paradigm for…
Forman-Ricci curvature (FRC) is a potent and powerful tool for analysing empirical networks, as the distribution of the curvature values can identify structural information that is not readily detected by other geometrical methods.…
For $X$ a metric space and $r>0$, the Vietoris--Rips simplicial complex $\mathrm{VR}(X;r)$ has $X$ as its vertex set, and a finite subset $\sigma \subseteq X$ as a simplex whenever the diameter of $\sigma$ is less than $r$. In ``On…
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…
We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let $\mathcal{U}$ be a cover of a separable metric space $X$ by open sets with a uniform diameter bound. The Vietoris…
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…
Persistent homology has been studied to better understand the structural properties and topology features of weighted networks. It can reveal hidden layers of information about the higher-order structures formed by non-pairwise interactions…
Topological Data Analysis (TDA) studies the shape of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukeherjee, and Boyer showed that…