Related papers: Sparips
Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…
Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…
Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…
Machine learning pipelines for classification tasks often train a universal model to achieve accuracy across a broad range of classes. However, a typical user encounters only a limited selection of classes regularly. This disparity provides…
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…
In this paper we define, implement, and investigate a simplicial complex construction for computing persistent homology of Euclidean point cloud data, which we call the Delaunay-Rips complex (DR). Assigning the Vietoris-Rips weights to…
Topological correctness is critical for segmentation of tubular structures, which pervade in biomedical images. Existing topological segmentation loss functions are primarily based on the persistent homology of the image. They match the…
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…
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,…
With the emergence of graph databases, the task of frequent subgraph discovery has been extensively addressed. Although the proposed approaches in the literature have made this task feasible, the number of discovered frequent subgraphs is…
In topology inference from data, current approaches face two major problems. One concerns the selection of a correct parameter to build an appropriate complex on top of the data points; the other involves with the typical `large' size of…
Persistent homology is a method from computational algebraic topology that can be used to study the "shape" of data. We illustrate two filtrations --- the weight rank clique filtration and the Vietoris--Rips (VR) filtration --- that are…
A topological approach to stratification learning is developed for point cloud data drawn from a stratified space. Given such data, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a…
Computational topologists recently developed a method, called persistent homology to analyze data presented in terms of similarity or dissimilarity. Indeed, persistent homology studies the evolution of topological features in terms of a…
Characterizing the structural properties of neural networks is crucial yet poorly understood, and there are no well-established similarity measures between networks. In this work, we observe that neural networks can be represented as…
Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and…
Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…
Topological features play an essential role in ensuring geometric plausibility and structural consistency in image analysis tasks such as segmentation and skeletonization. However, integrating topology-preserving learning based on simple…
We construct worst-case examples for the standard reduction algorithm for computing persistent homology. Our constructions are similar to the worst-case examples introduced by Morozov, but we replace the single-triangle arrangement with a…
Solving optimization tasks based on functions and losses with a topological flavor is a very active, growing field of research in data science and Topological Data Analysis, with applications in non-convex optimization, statistics and…