Topology and data

Carlsson, Gunnar

doi:10.1090/s0273-0979-09-01249-x

Cited by 1,750 publications

(1,532 citation statements)

References 50 publications

Supporting

Mentioning

1,449

Contrasting

Unclassified

Order By: Relevance

“…However, the morphology of an intermittent random field is only one of its aspects. More subtle but no less essential features are revealed by topological filtration, which characterises statistical properties of the extrema of the random field and connectivity of its isosurfaces (Carlsson 2009;Adler et al 2010;Adler & Taylor 2011;Edelsbrunner 2014). These features are described in terms of the Betti numbers, β 0 , β 1 and β 2 ; in a space of a dimension d, there are d Betti numbers.…”

Section: Introductionmentioning

confidence: 99%

Topological signatures of interstellar magnetic fields – I. Betti numbers and persistence diagrams

Makarenko

Shukurov

Henderson

et al. 2018

Monthly Notices of the Royal Astronomical Society

View full text Add to dashboard Cite

The interstellar medium (ISM) is a magnetised system in which transonic or supersonic turbulence is driven by supernova explosions. This leads to the production of intermittent, filamentary structures in the ISM gas density, whilst the associated dynamo action also produces intermittent magnetic fields. The traditional theory of random functions, restricted to second-order statistical moments (or power spectra), does not adequately describe such systems. We apply topological data analysis (TDA), sensitive to all statistical moments and independent of the assumption of Gaussian statistics, to the gas density fluctuations in a magnetohydrodynamic (MHD) simulation of the multi-phase ISM. This simulation admits dynamo action, so produces physically realistic magnetic fields. The topology of the gas distribution, with and without magnetic fields, is quantified in terms of Betti numbers and persistence diagrams. Like the more standard correlation analysis, TDA shows that the ISM gas density is sensitive to the presence of magnetic fields. However, TDA gives us important additional information that cannot be obtained from correlation functions. In particular, the Betti numbers per correlation cell are shown to be physically informative. Magnetic fields make the ISM more homogeneous, reducing the abundance of both isolated gas clouds and cavities, with a stronger effect on the cavities. Remarkably, the modification of the gas distribution by magnetic fields is captured by the Betti numbers even in regions more than 300 pc from the midplane, where the magnetic field is weaker and correlation analysis fails to detect any signatures of magnetic effects.

show abstract

Section: Introductionmentioning

confidence: 99%

Topological signatures of interstellar magnetic fields – I. Betti numbers and persistence diagrams

Makarenko

Shukurov

Henderson

et al. 2018

Monthly Notices of the Royal Astronomical Society

View full text Add to dashboard Cite

show abstract

“…Topological-based methods have certain properties that are not available for the geometrical-based methods [37]:…”

Section: Characteristics Of Geometric-based Data Analysis Methodsmentioning

confidence: 99%

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

2018

View full text Add to dashboard Cite

Atom probe tomography (APT) represents a revolutionary characterization tool for materials that combine atomic imaging with a time-of-flight (TOF) mass spectrometer to provide direct space three-dimensional, atomic scale resolution images of materials with the chemical identities of hundreds of millions of atoms. It involves the controlled removal of atoms from a specimen's surface by field evaporation and then sequentially analyzing them with a position sensitive detector and TOF mass spectrometer. A paradox in APT is that while on the one hand, it provides an unprecedented level of imaging resolution in three dimensions, it is very difficult to obtain an accurate perspective of morphology or shape outlined by atoms of similar chemistry and microstructure. The origins of this problem are numerous, including incomplete detection of atoms and the complexity of the evaporation fields of atoms at or near interfaces. Hence, unlike scattering techniques such as electron microscopy, interfaces appear diffused, not sharp. This, in turn, makes it challenging to visualize and quantitatively interpret the microstructure at the "meso" scale, where one is interested in the shape and form of the interfaces and their associated chemical gradients. It is here that the application of informatics at the nanoscale and statistical learning methods plays a critical role in both defining the level of uncertainty and helping to make quantitative, statistically objective interpretations where heuristics often dominate. In this chapter, we show how the tools of Topological Data Analysis provide a new and powerful tool in the field of nanoinformatics for materials characterization.

show abstract

“…The flag complex, as well as the so-called Vietoris-Rips complex (see [5]), is a standard tool used to perform topological data analysis [3]. In this section we describe an efficient bottom-up technique to obtain flag complexes.…”

Section: Construction Of Flag Complexmentioning

confidence: 99%

“…Still, we believe that experimenting with smaller, properly sampled data can give interesting insights. For example, [3] shows that data coming from natural images form a topological Klein bottle.…”

Section: Introduction and Existing Workmentioning

confidence: 99%

See 1 more Smart Citation

Computational Topology in Text Mining

Wagner

Dłotko

Mrożek

2012

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. In this paper we present our ongoing research on applying computational topology to analysis of structure of similarities within a collection of text documents. Our work is on the fringe between text mining and computational topology, and we describe techniques from each of these disciplines. We transform text documents to the so-called vector space model, which is often used in text mining. This representation is suitable for topological computations. We compute homology, using Discrete Morse theory, and persistent homology of the Flag complex built from the point-cloud representing the input data. Since the space is highdimensional, many difficulties appear. We describe how we tackle these problems and point out what challenges are still to be solved.

show abstract

Topology and data

Cited by 1,750 publications

References 50 publications

Topological signatures of interstellar magnetic fields – I. Betti numbers and persistence diagrams

Topological signatures of interstellar magnetic fields – I. Betti numbers and persistence diagrams

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

Computational Topology in Text Mining

Contact Info

Product

Resources

About