Utility of Betti Sequences as Persistent Homology-based Topological Descriptors in Application to Inference for Space-Time Processes and Time Series of Complex Networks

Everything we see around us has a shape. In the age when data are being collected at unrepresented rates and volumes, we often come across data types having inherent shape structure. Rapidly developing field of topological data analysis (TDA) offers rigorous mathematical methods and efficient computational tools to describe and analyze the shape structure of data. One of the advantages of the topological approach is that it is more robust against the choice of a distance metric than geometric methods by being coordinate free. In this dissertation, we present new approaches and perspectives of applying TDA concepts in the contexts of various statistical and learning tasks such as unsupervised clustering, anomaly detection in complex networks, time series change point detection and non-rigid 3D shape analysis. More specifically, we explore the properties and utility of the so-called Betti sequences as topological descriptors or summaries extracted from data. Under the TDA approach to data analysis, such topological summaries are often extracted via a medium of combinatorial and topological structures built on top of data called abstract simplicial complexes. Choosing an appropriate type of a simplicial complex for a given dataset and learning task depends on computational complexity of the problem as well as which features of the data one expects to highlight. We investigate this interplay between various types of simplicial complexes and specific learning frameworks under which they are used. We also contrast the Betti sequence with other topological summaries such as persistence diagrams and point out their advantages in terms of computational cost and ease of incorporation into relevant learning settings for further analysis of data.