Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTopological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers
156
0
0.0
(
0
)
Added by
Tamal Dey
Publication date
2017
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called emph{nerve} of a cover of the domain. In this paper, we aim to analyze the topological information encoded in these structures in order to provide better understanding of these structures and facilitate their practical usage. More specifically, we show that the one-dimensional homology of the nerve complex $N(mathcal{U})$ of a path-connected cover $mathcal{U}$ of a domain $X$ cannot be richer than that of the domain $X$ itself. Intuitively, this result means that no new $H_1$-homology class can be created under a natural map from $X$ to the nerve complex $N(mathcal{U})$. Equipping $X$ with a pseudometric $d$, we further refine this result and characterize the classes of $H_1(X)$ that may survive in the nerve complex using the notion of emph{size} of the covering elements in $mathcal{U}$. These fundamental results about nerve complexes then lead to an analysis of the $H_1$-homology of Reeb spaces, mappers and multiscale mappers. The analysis of $H_1$-homology groups unfortunately does not extend to higher dimensions. Nevertheless, by using a map-induced metric, establishing a Gromov-Hausdorff convergence result between mappers and the domain, and interleaving relevant modules, we can still analyze the persistent homology groups of (multiscale) mappers to establish a connection to Reeb spaces.
rate research
Read More
The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it compresses the components of the level sets of a multivariate mapping and obtains a summary representation of their relationships. A related construction called mapper, and a special case of the mapper construction called the Joint Contour Net have been shown to be effective in visual analytics. Mapper and JCN are intuitively regarded as discrete approximations of the Reeb space, however without formal proofs or approximation guarantees. An open question has been proposed by Dey et al. as to whether the mapper construction converges to the Reeb space in the limit. In this paper, we are interested in developing the theoretical understanding of the relationship between the Reeb space and its discrete approximations to support its use in practical data analysis. Using tools from category theory, we formally prove the convergence between the Reeb space and mapper in terms of an interleaving distance between their categorical representations. Given a sequence of refined discretizations, we prove that these approximations converge to the Reeb space in the interleaving distance; this also helps to quantify the approximation quality of the discretization at a fixed resolution.
In this article, we show how the recent statistical techniques developed in Topological Data Analysis for the Mapper algorithm can be extended and leveraged to formally define and statistically quantify the presence of topological structures coming from biological phenomena in datasets of CCC contact maps.
Almost all statistical and machine learning methods in analyzing brain networks rely on distances and loss functions, which are mostly Euclidean or matrix norms. The Euclidean or matrix distances may fail to capture underlying subtle topological differences in brain networks. Further, Euclidean distances are sensitive to outliers. A few extreme edge weights may severely affect the distance. Thus it is necessary to use distances and loss functions that recognize topology of data. In this review paper, we survey various topological distance and loss functions from topological data analysis (TDA) and persistent homology that can be used in brain network analysis more effectively. Although there are many recent brain imaging studies that are based on TDA methods, possibly due to the lack of method awareness, TDA has not taken as the mainstream tool in brain imaging field yet. The main purpose of this paper is provide the relevant technical survey of these powerful tools that are immediately applicable to brain network data.
Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under $mathbb{Z}_2$ coefficients for a sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis. First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provides an algorithm for computing the persistence induced by simplicial maps. Our main result is that the above simple minded approach can be improved for a sequence of simplicial maps given in a monotone direction. A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses--two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally. Finally, we exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.
Reeb graphs are structural descriptors that capture shape properties of a topological space from the perspective of a chosen function. In this work we define a combinatorial metric for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations. The main contributions of this paper are the stability property and the optimality of this edit distance. More precisely, the stability result states that changes in the functions, measured by the maximum norm, imply not greater changes in the corresponding Reeb graphs, measured by the edit distance. The optimality result states that our edit distance discriminates Reeb graphs better than any other metric for Reeb graphs of surfaces satisfying the stability property.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
