Typed Topological Structures Of Datasets

TL;DR

This paper introduces a novel typed topological framework for datasets in R^2, enabling detailed structural analysis and new algorithms for geometric and clustering problems.

Contribution

It develops a specific set of types and related topology to analyze dataset structures, including tracks, components, and pseudotrees, offering new tools for data analysis.

Findings

Organizes dataset into tracks and components with an order

Represents relationships using typed-II pseudotrees

Provides a platform for algorithms on convex hulls, holes, clustering, and anomalies

Abstract

A datatset $X$ on $R^2$ is a finite topological space. Current research of a dataset focuses on statistical methods and the algebraic topological method \cite{carlsson}. In \cite{hu}, the concept of typed topological space was introduced and showed to have the potential for studying finite topological spaces, such as a dataset. It is a new method from the general topology perspective. A typed topological space is a topological space whose open sets are assigned types. Topological concepts and methods can be redefined using open sets of certain types. In this article, we develop a special set of types and its related typed topology on a dataset $X$. Using it, we can investigate the inner structure of $X$. In particular, $R^2$ has a natural quotient space, in which $X$ is organized into tracks, and each track is split into components. Those components are in a order. Further, they can be represented by an integer sequence. Components crossing tracks form branches, and the relationship can be well represented by a type of pseudotree (called typed-II pseudotree). Such structures provide a platform for new algorithms for problems such as calculating convex hull, holes, clustering and anomaly detection.