Applications of Algebraic Topology in Data Analysis

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Gol Mohammadian Tehrani, Mohammad Hassan | 2019

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 52205 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Jafari, Amir; Parsa, Salman
Abstract:
The goal of this thesis is to represent some application of algebraic topology specially homology and simplicial structures in data analysis and image processing. First the notion of topological persistence will be introduced and next for computing that several efficient algorithms will be proposed. Also complexity of the algorithms base on the topological spaces and by some algebraic topology tools. And it will be tried to test this theorical algorithms on real data and represent some application of topology in real data analysis. To do that, at the begining the tools of algebraic topology will be reviewed and the notions like singular complexes and singular homology and some related property of them. Furthermore, using
topological persistence, the notion of simplification of geometric filtrations and some application
of that will be stated. It can be mentioned some other application in data analysis like
compression of data using geometric filtrations
Keywords:
Alpha Shapes ; Computational Algebraic Topology ; Data Analysis ; Image Processing ; Homology ; Alpha Shapes