A new compressed cover tree for k-nearest neighbour search and the stable-under-noise mergegram of a point cloud

TL;DR

This thesis introduces a new compressed cover tree for efficient k-nearest neighbor search, a method for constructing MSTs in metric spaces, and extends persistence concepts in TDA to analyze data topology.

Contribution

It presents a novel compressed cover tree data structure, a generalized MST construction method, and an extension of persistence in TDA, combining computational geometry and algebraic topology.

Findings

Improved efficiency in k-nearest neighbor search using the new cover tree.

A generalized method for constructing MSTs in any finite metric space.

Extension of persistence concepts to new topological data analysis applications.

Abstract

This thesis consists of two topics related to computational geometry and one topic related to topological data analysis (TDA), which combines fields of computational geometry and algebraic topology for analyzing data. The first part studies the classical problem of finding k nearest neighbors to m query points in a larger set of n reference points in any metric space. The second part is about the construction of a Minimum Spanning Tree (MST) on any finite metric space. The third part extends the key concept of persistence within Topological Data Analysis in a new direction.