Barcodes: The persistent topology of data

By Robert Ghrist. 2007

1,378 citations recorded by the source corpus.

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets — persistent homology — and a novel representation of this algebraic characterization — barcodes. We sketch an application of these techniques to the classification of natural images. 1. The shape of data When a topologist is asked, “How do you visualize a four-dimensional object?” the appropriate response is a Socratic rejoinder: “How do you visualize a threedimensional object?” We do not see in three spatial dimensions directly, but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended. We spend a significant portion of our first year of life learning how to infer three-dimensional spatial data from paired planar projections. Years of practice have tuned a remarkable ability to extract global structure from representations in a strictly lower dimension. The inference of global structure occurs on much finer scales as well, with regards to converting discrete data into continuous images. Dot-matrix printers, scrolling LED tickers, televisions, and computer displays all communicate images via arrays of discrete points which are integrated into coherent, global objects. This also is a skill we have practiced from childhood. No adult does a dot-to-dot puzzle with anything approaching anticipation. 1.1. Topological data analysis. Problems of data analysis share many features with these two fundamental integration tasks: (1) how does one infer high dimensional structure from low dimensional representations; and (2) how does one assemble discrete points into global structure. The principal themes of this survey of the work of Carlsson, de Silva, Edelsbrunner, Harer, Zomorodian, and others are the following: (1) It is beneficial to replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. This converts the data set into global topological objects. (2) It is beneficial to view these topological complexes through the lens of algebraic topology — specifically, via a novel theory of persistent homology adapted to parameterized families. (3) It is beneficial to encode the persistent homology of a data set in the form of a parameterized version of a Betti number: a barcode. The author gratefully acknowledges the support of DARPA # HR0011-07-1-0002. The work reviewed in this article is funded by the DARPA program TDA: Topological Data Analysis.

Subfields: Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cad0bff789bb0e626245b21a47791ef76a9cd5c2

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Citations in the corpus

Statistical topological data analysis using persistence landscapes (2012, background): “Of particular interest are the homological features that persist as the resolution changes.”
Weighted persistent homology (2017, neutral): “First, we give a review of persistence [14, 25], with generalizations to the weighted case.”
Topological complexity of smooth random functions (2011, background): “A very lively discussion of this trend can be found in Rob Ghrist’s review [40], book in progress [41] and website on a project on sensor topology for minimal planning [42].”
Persistence modules with operators in Morse and Floer theory (2017, background): “For detailed treatment of these topics see [8, 9, 12,17,34,62].”
Topology of random simplicial complexes: a survey (2013, background): “1. Motivation. 1.1.1. Randomnessmodelsthenaturalworld. (1)Theﬁeldoftopologicaldataanalysishasreceivedalotofattentionover thepastseveralyears—see,forexample,thesurveyarticlesofCarlsson [16] and Ghrist [30]. In order to quantify the statistical signiﬁcance of topologicalfeaturesdetectedwiththesemethod”
Betti numbers in multidimensional persistent homology are stable functions (2013, neutral): “Recent surveys on the topic include [19, 20, 26, 35].”
Topological Data Analysis (2012, background): “Recent surveys on topological analysis include Ghrist [35] and Carlsson [5].”
Closures and Cavities in the Human Connectome (2016, methodology): “In order to quantitatively characterize this macroscale structure, we employ an enrichment of networks that comes from the field of algebraic topology [26], developed precisely to understand the interplay between these weak and strong connections in systems [25].”
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.”
Persistence Images: A Stable Vector Representation of Persistent Homology (2015, background): “The idea of persistent homology is to compute homology at many scales and observe which topological features persist across those scales (Ghrist, 2008; Carlsson, 2009; Edelsbrunner and Harer, 2008).”
Bounds on spectral norms and barcodes (2018, background): “ersistence modules and boundary depth. We recall brie y the category pmod of persistence modules that we work with, together with their relevant properties. For detailed treatment of these topics see [12, 22, 23, 34, 49, 121]. Let K be a eld. A persistence module over K is a pair (V;ˇ) where, fVtg t”
Decomposition of pointwise finite-dimensional persistence modules (2012, background): “) Assuming that this persistence module is a direct sum of interval modules, it is found that the collection of such intervals, the barcode, is a useful invariant of the data; see for example [1, 6].”
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”
A Complete Characterization of the 1-Dimensional Intrinsic Cech Persistence Diagrams for Metric Graphs (2017, methodology): “For a more detailed description of applications of persistent homology to various problems in the experimental sciences, see [5, 8, 11].”