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

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 well-known work by Beygelzimer, Kakade, and Langford in ICML 2006 introduced cover trees and claimed to guarantee a nearlinear time complexity in the number n of reference points. Section 5.3 of Curtin's PhD (2015) pointed out that the proof of this result was wrong. The key step of the original proof attempted to show that the number of iterations can be estimated by multiplying the length of the longest rootto-leaf path in a cover tree by a constant factor. However, this estimate can miss many potential nodes in several branches of a cover tree, that should be considered during the search. The same erroneous argument was unfortunately repeated in several subsequent papers. This chapter gives formal counterexamples to the time complexity estimates of the cover tree construction algorithm and the nearest neighbors search in the past work. We prove correct analogs of the past claims with slightly better bounds for any number k ≥ 1 of neighbors. A new compressed cover tree guarantees a parameterized time complexity that is near-linear in the maximum size of both query and reference set.The second part is about the construction of a Minimum Spanning Tree (MST) on any finite metric space. This is an efficient way to visualize any unstructured data given only by distances, for example by any metric graph connecting abstract data points. Though many efficient algorithms for MST were developed, there was only one past attempt to justify a nearlinear time complexity in the size of a given metric space. In 2010 March, Ram, and Gray claimed that MST of any finite metric space can be constructed in a parametrized near-linear time. However, in this work we have discovered a counterexample showing that the provided proof was incorrect. The past obstacles are now resolved by the new concepts of a compressed cover tree and a minimized expansion constant. The main contribution is the first MST algorithm that correctly justifies a parametrized a near-linear time complexity.The third part extends the key concept of persistence within Topological Data Analysis in a new direction. TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0-dimensional case, the distance-based persistence is determined by a i First of all, I would like to thank my primary supervisor, Dr. Vitaliy Kurlin, for giving me the opportunity to pursue a Ph.D. in a field about which I am very passionate and for always supporting me during these years. His incredible experience has been inspiring to me. I am grateful for all our fruitful conversations that helped me progress in my research.I would also like to thank my second supervisor, Dr. Marja K...