相关论文: Graph Induced Complex on Point Data

This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can…

Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way…

Direct B-Rep generation is increasingly important in CAD workflows, eliminating costly modeling sequence data and supporting complex features. A key challenge is modeling joint distribution of the misaligned geometry and topology. Existing…

In a world abundant with diverse data arising from complex acquisition techniques, there is a growing need for new data analysis methods. In this paper we focus on high-dimensional data that are organized into several hierarchical datasets.…

In recent years, network embedding methods have garnered increasing attention because of their effectiveness in various information retrieval tasks. The goal is to learn low-dimensional representations of vertexes in an information network…

Capitalizing on the intuitive premise that shape characteristics are more robust to perturbations, we bridge adversarial graph learning with the emerging tools from computational topology, namely, persistent homology representations of…

We propose a graph semi-supervised learning framework for classification tasks on data manifolds. Motivated by the manifold hypothesis, we model data as points sampled from a low-dimensional manifold $\mathcal{M} \subset \mathbb{R}^F$. The…

This research uses deep learning to estimate the topology of manifolds represented by sparse, unordered point cloud scenes in 3D. A new labelled dataset was synthesised to train neural networks and evaluate their ability to estimate the…

We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980s with Vietoris-Rips Persistent Homology. For given integers $k\geq 0$ and $n\geq 1$ we consider the dimension $k$…

While the Vietoris-Rips complex is now widely used in both topological data analysis and the theory of hyperbolic groups, many of the fundamental properties of its homology have remained elusive. In this article, we define the Vietoris-Rips…

We derive conditions under which the reconstruction of a target space is topologically correct via the \v{C}ech complex or the Vietoris-Rips complex obtained from possibly noisy point cloud data. We provide two novel theoretical results.…

Graphs are widely used to represent complex information and signal domains with irregular support. Typically, the underlying graph topology is unknown and must be estimated from the available data. Common approaches assume pairwise node…

Topology identification and inference of processes evolving over graphs arise in timely applications involving brain, transportation, financial, power, as well as social and information networks. This chapter provides an overview of graph…

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…

We consider the problem of homotopy-type reconstruction of compact subsets $X\subset\R^N$ that have the Alexandrov curvature bounded above ($\leq$ $\kappa$) in the intrinsic length metric. The reconstructed spaces are in the form of…

Performing statistical analyses on collections of graphs is of import to many disciplines, but principled, scalable methods for multi-sample graph inference are few. Here we describe an "omnibus" embedding in which multiple graphs on the…

Sampling from distributions of implicitly defined shapes enables analysis of various energy functionals used for image segmentation. Recent work describes a computationally efficient Metropolis-Hastings method for accomplishing this task.…

Graphs are powerful abstractions for capturing complex relationships in diverse application settings. An active area of research focuses on theoretical models that define the generative mechanism of a graph. Yet given the complexity and…

Topological Data Analysis (TDA) has been praised by researchers for its ability to capture intricate shapes and structures within data. TDA is considered robust in handling noisy and high-dimensional datasets, and its interpretability is…

Large datasets represented by multidimensional data point clouds often possess non-trivial distributions with branching trajectories and excluded regions, with the recent single-cell transcriptomic studies of developing embryo being notable…