Persistent Homology and its Applications in Machine Learning

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Kiani, Amir | 2023

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 56686 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Ranjbar, Alireza
Abstract:
Persistent homology is one of the main tools in Topological Data Analysis. Indeed, to deal with a huge dataset while noise sensitivity is important, persistent homology can reflect some information about data in the form of persistent homology groups and persistence diagrams. Note that statistical or linear algebraic tools are not suitable to work with huge datasets with very high dimensions. In this thesis, we discuss the concept of persistent homology and investigate some of its properties such as the stability of the persistence diagrams. Indeed, persistence diagrams are obtained from the generating sets of the persistent homology groups. Further, we discuss an application of persistent homology in machine learning via the theory of group equivariant non-expansive operators. The latter is a topological-geometric framework to deal with machine learning
Keywords:
Persistent Homology ; Machine Learning ; Group Equivariant Operator ; Topological Data Analysis