טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentReani Yohai
SubjectCycle Registration in Persistent Homology with
Application in Topological Bootstrap
DepartmentDepartment of Electrical Engineering
Supervisor Dr. Omer Bobrowski
Full Thesis textFull thesis text - English Version


Abstract

Topological Data Analysis (TDA) refers to a set of methods used to analyze the “shape” of data using ideas from mathematical (mainly algebraic) topology. The “shape” of a set of data points, is concretely realized by imposing a structure known as simplicial complex (high-dimensional analogue of a graph) on these data points. One of the main objects used in TDA is homology, which characterize shapes by their connected components, “holes”, “voids”, and higher dimensional generalizations of these. Studying the homology of data is mostly done through its multiscale version, known as persistent homology, which provides a summary of the evolution of homological features as the scale in which we view the data varies. In many cases the need for comparing persistent homology representations arises. Commonly used comparison methods are based on numerical summaries such as persistence diagrams and persistence landscapes. These low dimensional summaries - of potentially high dimensional data, are useful for computational purposes. However, they ignore significant structural information intrinsic to the data. In this thesis we propose a novel approach for comparing the persistent homology representations of two spaces. We do so by defining a correspondence relation between individual persistent cycles of two different spaces, which is based on both the lifetime interval and the spatial placement of each cycle. This relation gives rise to the definition of a new variant of homology, which enables us to analyze the homology of a space from the perspective of a subspace. We supply algorithms for computing our matching method for the important case where the spaces under inspection are simplicial complexes generated by point cloud data. While our method has a natural implementation for the well-known Čech and Rips complexes, the use of these complexes may become computationally expensive. Therefore, we devise a new construction based on the alpha complex, that is of a significantly smaller size, and therefore, yields faster homology calculations. Finally, we apply our new framework in topological inference, addressing the problem of differentiating real features from noise in point cloud data. In particular, our new framework facilitates the use of statistical bootstrap to assess the significance of cycles.