Short Communication, Res Rep Math Vol: 8 Issue: 1
A Survey of Topological Data Analysis and its Applications
Alberta Gorge*
1Department of Mathematics, University of Toronto, Toronto, Ontario, Canada
*Corresponding Author: Alberta Gorge
Department of Mathematics, University of
Toronto, Toronto, Ontario, Canada
E-mail: gorge@alber.ca
Received date: 02 February, 2023, Manuscript No. RRM-23-92317;
Editor assigned date: 06 February, 2023, Pre QC No. RRM-23-92317(PQ);
Reviewed date: 13 February, 2023, QC No. RRM-23-92317;
Revised date: 20 February, 2023, Manuscript No. RRM-23-92317(R);
Published date: 27 February, 2023, DOI: 0.4172/rrm.1000173
Citation: Gorge A (2023) A Survey of Topological Data Analysis and its Applications. Res Rep Math 7:1.
Description
Topological Data Analysis (TDA) is an emerging field that combines algebraic topology and computational geometry to study the shape and structure of complex datasets. TDA has gained significant attention in recent years due to its ability to extract meaningful information from high-dimensional and noisy data.
A topological space is a set of points equipped with a notion of proximity, called a topology. Intuitively, two points are considered close if they can be connected by a continuous path without passing through any obstacles. Algebraic topology provides a toolkit for analyzing the topological properties of these spaces. In particular, the homology groups of a topological space capture its connectivity and holes [1]. Homology groups are algebraic objects that encode the number and dimensionality of topological features of a space. The key idea in TDA is to use these topological invariants to analyze the structure of datasets. Given a dataset, one can construct a simplicial complex by connecting points based on their proximity. The resulting complex encodes the topological features of the data, such as clusters, holes, and voids. One can then compute the homology groups of this complex to extract information about the data structure. In particular, the zeroth homology group (the number of connected components) reveals the number of clusters in the data, while the first homology group (the number of one-dimensional holes) captures the presence of voids [2].
There are several methods used in TDA to compute the homology groups of a dataset. One approach is persistent homology, which tracks the evolution of the homology groups as the scale of the data is varied. This provides a robust and stable way to extract topological information from noisy and high-dimensional data [3].
TDA has found numerous applications in various fields, including neuroscience, biology, materials science, and computer vision. TDA has been used to study the structure of brain networks and to identify biomarkers for Alzheimer's disease. In materials science, TDA has been used to classify and predict the properties of materials based on their atomic structures. In computer vision, TDA has been used to analyze the shape and structure of images and to classify objects based on their topological features [4].
Application of Topological Data Analysis (TDA)
One of the most prominent applications of topology is in geometry and topology of manifolds. A manifold is a topological space that is locally Euclidean, meaning that it looks like Euclidean space in small regions. Topology provides tools for studying the properties of manifolds, such as their dimensions, curvature, and topology. The topology of manifolds has applications in geometry, physics, and engineering, where it is used to study the properties of surfaces, spaces, and materials [5].
Conclusion
Topological Data Analysis (TDA) is a powerful tool for analyzing complex and high-dimensional data sets. TDA provides a framework for identifying the topological features and patterns in data that are not readily apparent through traditional statistical techniques. TDA has applications in a wide range of fields, including biology, genetics, neuroscience, materials science, machine learning, social network analysis, urban planning, and finance. The continued development of TDA is likely to lead to further insights and innovations in these fields.
References
- Larry W (2018) Topological data analysis. Annu Rev Stat Appl 5 501-532.
- Peter B (2015) Statistical topological data analysis using persistence landscapes. J Mach Learn Res 16(1): 77-102. .
- Chazal (2021) An introduction to topological data analysis: fundamental and practical aspects for data scientists. Front Artif Intell 4(1): 667963. .
- Topaz (2015) Topological data analysis of biological aggregation models. Plos one 10(5): 152-157. .
- Skaf (2022) Topological data analysis in biomedicine: A Review. J Biomed Inform 130.
