Topology and data

By Gunnar E. Carlsson. 2009

2,658 citations recorded by the source corpus.

An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data we are obtaining is significantly different. For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones. A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it. The data obtained is also often much noisier than in the past and has more missing information (missing data). This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources. Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced. In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data. Geometry and topology are very natural tools to apply in this direction, since geometry can be regarded as the study of distance functions, and what one often works with are distance functions on large finite sets of data. The mathematical formalism which has been developed for incorporating geometric and topological techniques deals with point clouds, i.e. finite sets of points equipped with a distance function. It then adapts tools from the various branches of geometry to the study of point clouds. The point clouds are intended to be thought of as finite samples taken from a geometric object, perhaps with noise. Here are some of the key points which come up when applying these geometric methods to data analysis. • Qualitative information is needed: One important goal of data analysis is to allow the user to obtain knowledge about the data, i.e. to understand how it is organized on a large scale. For example, if we imagine that we are looking at a data set constructed somehow from diabetes patients, it would be important to develop the understanding that there are two types of the disease, namely the juvenile and adult onset forms. Once that is established, one of course wants to develop quantitative methods for distinguishing them, but the first insight about the distinct forms of the disease is key.

Subfields: Biophysics, Computational Theory and Mathematics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/a4b603ca6aaaa18968e08ac1b0ee093db8a99a6b

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Citations in the corpus

Cliques and cavities in the human connectome (2016, methodology): “First we record cycle lifetime l = ρdeath − ρbirth, which is commonly used in persistent homology calculations (Carlsson 2009) and displayed on a persistence diagram.”
Accelerated Hierarchical Density Based Clustering (2017, methodology): “is is persistent homology [17], [64]. Although our description of HDBSCAN* makes use of persistent homology techniques, the full details of that subject are beyond the scope of this paper. Please see [8] and [22] for a good introduction to the topic. We will also make use of the language of sheaves.”
Statistical topological data analysis using persistence landscapes (2012, background): “tleneck and Wasserstein distances. 1. Introduction Topological data analysis (TDA) consists of a growing set of methods that provide insight to the \shape" of data (see the surveys Ghrist, 2008; Carlsson, 2009). These tools may be of particular use in understanding global features of high dimen”
Persistence modules with operators in Morse and Floer theory (2017, background): “For detailed treatment of these topics see [8, 9, 12,17,34,62].”
A Complete Characterization of the 1-Dimensional Intrinsic Cech Persistence Diagrams for Metric Graphs (2017, background): “Persistent homology [5, 10] then tracks elements of each homology group through the filtration.”
Persistent homology for random fields and complexes (2010, background): “Recent excellent and quite different reviews by Carlsson [9, 12] Edelsbrunner [19] and Ghrist [1, 16, 17, 22] give a broad exposition of the basics of persistent homology.”
Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules (2014, methodology): “ on the homology level does not depend on the choice of almost complex structures, the lemma now follows immediately. 4 Invariants 4.1 The Zk spectral spread In the spirit of persistent homology (see [49, 25, 11] for surveys), a rapidly developing area lying on the borderline of algebraic topology a”
Random geometric complexes in the thermodynamic regime (2014, background): “rest in random geometric complexes was stimulated by their connections to topological data analysis and, more broadly, applied topology. There are a number of accessible surveys on this subject (e.g. [6, 9, 12, 15, 45]), all of which share a common theme of studying topological invariants of simplic”
Betti numbers in multidimensional persistent homology are stable functions (2013, background): “This research area is widely discussed in [2, 4].”
What are higher-order networks? (2021, background): “When data is given as a point cloud, one needs to determine what is the appropriate filtration parameter and corresponding complex to build [51, 87, 88, 110].”
Morse Theory for Filtrations and Efficient Computation of Persistent Homology (2013, methodology): “The use of topological methods for data analysis is rapidly growing and persistent homology is proving to be one of the more successful techniques [1, 7, 8, 11].”
Maximally Persistent Cycles in Random Geometric Complexes (2015, background): “Commonly (see [16, 33]), the persistence of a cycle is measured by the length of the corresponding bar in the barcode, namely by the difference δ(γ ) := γdeath − γbirth.”
Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival (2011, background): “The fundamental mathematical challenges in extracting meaningful information from high-throughput biological data stem, ultimately, from the difficulty in understanding the intrinsic shape of data in high dimensions (1).”
Random Walks on Simplicial Complexes and the normalized Hodge Laplacian (2018, background): “m walks on graphs and the graph Laplacian to develop a random walk on simplicial complexes that relates to the Hodge Laplacian. Furthermore, the connection of simplicial complexes to applied topology [21,35] has already led to success in analyzing biological data [23,55], structural and functional c”
On the topology of random complexes built over stationary point processes. (2012, background): “s of the Vietoris–Rips complex. 1.3. Some implications for topological data analysis. Perhaps the core tool of TDA is persistent homology, as visualized through barcodes and persistence diagrams; cf. [11, 15, 19, 42]. See also the very accessible recent survey [12]. While here is not the place to go”
Deep Learning with Topological Signatures (2017, background): “Several works have demonstrated that TDA can be beneﬁcial in a diverse set of problems, such as studying the manifold of natural image patches [7], analyzing activity patterns of the visual cortex [26], classiﬁcation of 3D surface meshes [25, 20], clustering [10], or recognition of 2D object shapes ”
Closures and Cavities in the Human Connectome (2016, methodology): “First we record cycle lifetime l = ρdeath − ρbirth, which is commonly used in persistent homology calculations [13] and displayed on a persistence diagram.”
Classifying Clustering Schemes (2010, methodology): “It is also true that functoriality is necessary to extend the methods of clustering to higher order connectivity information with computational topology of ﬁnite datasets [3].”