Configuration spaces of robotic hands

In this thesis we extend topological model of planar robotic hands emerging in the field of topological robotics. This research elaborates further recent works of Robert Ghrist and others. The main purpose of this thesis is to classify configuration spaces in terms of topological and algebraic invariants, which among others provides complexity estimates for potential optimization algorithms. The thesis is split into two parts. In the first part we investigate a robotic system consisting of a single hand which can occupy any position as long as it doesn't self-intersect. Using a new innovative representation of positions we are able to treat two basic movements of the robotic arm: the 'claw' and the 'swap' movements separately. The main appliance of this part is the nerve theorem, which helps to establish that under some restrictions the configuration space of such robotic hand has the homotopy type of S^1. In the second part we investigate systems consisting of multiple hands. This time we are dealing with hands limited to length one whose positions satisfy the two conditions: each pairwise hand trace intersection is contractible and the hand intersection graph is a forest. As the local main result we prove that the fundamental group of such robotic system is isomorphic to the Artin right-angeled group, where the set of generators is in bijection with the set of all hands and relations are determined by the intersection graph. The main tool exploited in this chapter is the Seifert-van Kampen theorem. Although the results are proven only for some special cases, the thesis introduces methodology that can drive their generalization further. In the final chapter we give a few sophisticated research directions

A fast approximate skeleton with guarantees for any cloud of points in a Euclidean space

The tree reconstruction problem is to find an embedded straight-line tree that approximates a given cloud of unorganized points in $\mathbb{R}^m$ up to a certain error. A practical solution to this problem will accelerate a discovery of new colloidal products with desired physical properties such as viscosity. We define the Approximate Skeleton of any finite point cloud $C$ in a Euclidean space with theoretical guarantees. The Approximate Skeleton ASk$(C)$ always belongs to a given offset of $C$, i.e. the maximum distance from $C$ to ASk$(C)$ can be a given maximum error. The number of vertices in the Approximate Skeleton is close to the minimum number in an optimal tree by factor 2. The new Approximate Skeleton of any unorganized point cloud $C$ is computed in a near linear time in the number of points in $C$. Finally, the Approximate Skeleton outperforms past skeletonization algorithms on the size and accuracy of reconstruction for a large dataset of real micelles and random clouds

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

The analysis of data sets mathematically representable as finite metric spaces plays a significant role in every scientific study. In this thesis we focus on constructing new effective algorithms in the area of computational geometry that can be effectively deployed for the study and classification of large data sets prevalent in natural science, economic analysis, medicine, environmental protection etc. The first major contribution of this thesis is a new near-linear time algorithm, that resolves the classical problem of finding $k$-nearest neighbors (KNN) to of query set $Q$ in a larger reference set $R$, where $Q$ and $R$ both belong to some metric space $X$. This project was inspired by the work of Beygelzimer, Kakade, and Langford in ICML 2006 that attempted to show that such problem is resolvable for $k=1$ having a near-linear time complexity. However, in 2015 it was pointed out that the proof of their time complexity might contain mistakes, which has been ascertained in this thesis by showing that the proposed proof does not withstand a concrete counterexample. An important application of the KNN algorithm is a KNN graph on a finite metric space $R$ whose edge set is formed by connecting every point $p \in R$ with its $k$-nearest neighbors. The KNN graph finds its application in areas of data-skeletonization, where it can serve as an initial skeleton of the data set, or in cluster analysis, where connected components of the KNN graph can represent the clusters. Another application of the the KNN algorithm is Minimum spanning tree (MST), which is an efficient way to visualize any unstructured data while knowing only distances, for example any metric graph connecting abstract data points. Although many efficient algorithms for the MST in metric spaces have been devised, there existed only one past attempt to justify a near-linear 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 built in a parametrized near-linear time. In this work we have demonstrated, with multiple counterexamples, that the attempted proof was incorrect by showing that one of its step fails. Encouraged by the results of the work of 2010 this thesis produces a new algorithm that is based on Boruvka algorithm, which is combined with the KNN method to resolve the metric MST problem in a near-linear time complexity. In the thesis final chapter the MST algorithm is applied in the computation of a new isometry invariant mergegram of Topological Data Analysis (TDA). TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the $0$-dimensional ($0$D) case, the distance-based persistence is determined by a single-linkage (SL) clustering of a finite set in a metric space. Equivalently, the $0D$ persistence captures only edge lengths of a Minimum Spanning Tree (MST). Both the SL dendrogram and the MST are unstable under perturbations of points. In this thesis, we define the new stable-under-noise mergegram which outperforms previous isometry invariants on a classification of point clouds. In conclusion, the developed fast algorithms of this thesis can cater to a vast varieties of tasks in data science and beyond. The newly proposed corrected time complexity analysis of KNN and MST not only rectifies the past issues in their theoretical justifications but also gives a way to fix analogous issues in other similar methods based on the cover tree data structure

Paired compressed cover trees guarantee a near linear parametrized complexity for all kk-nearest neighbors search in an arbitrary metric space

This paper studies the important problem of finding all $k$-nearest neighbors to points of a query set $Q$ in another reference set $R$ within any metric space. Our previous work defined compressed cover trees and corrected the key arguments in several past papers for challenging datasets. In 2009 Ram, Lee, March, and Gray attempted to improve the time complexity by using pairs of cover trees on the query and reference sets. In 2015 Curtin with the above co-authors used extra parameters to finally prove a time complexity for $k=1$. The current work fills all previous gaps and improves the nearest neighbor search based on pairs of new compressed cover trees. The novel imbalance parameter of paired trees allowed us to prove a better time complexity for any number of neighbors $k\geq 1$

Counterexamples expose gaps in the proof of time complexity for cover trees introduced in 2006

This paper is motivated by the k-nearest neighbors search: given an arbitrary metric space, and its finite subsets (a reference set R and a query set Q), design a fast algorithm to find all k-nearest neighbors in R for every point q ∈ Q. In 2006, Beygelzimer, Kakade, and Langford introduced cover trees to justify a near-linear time complexity for the neighbor search in the sizes of Q,R.Section 5.3 of Curtin's PhD (2015) pointed out that the proof of this result was wrong. The key step in the original proof attempted to show that the number of iterations can be estimated by multiplying the length of the longest root-to-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 neighbor search. The same argument was unfortunately repeated in several subsequent papers using cover trees from 2006.This paper explicitly constructs challenging datasets that provide counterexamples to the past proofs of time complexity for the cover tree construction, the k-nearest neighbor search presented at ICML 2006, and the dual-tree search algorithm published in NIPS 2009.The corrected near-linear time complexities with extra parameters are proved in another forthcoming paper by using a new compressed cover tree simplifying the original tree structure

Counterexamples expose gaps in the proof of time complexity for cover trees introduced in 2006

This paper is motivated by the k-nearest neighbors search: given an arbitrary metric space, and its finite subsets (a reference set R and a query set Q), design a fast algorithm to find all k-nearest neighbors in R for every point q in Q. In 2006, Beygelzimer, Kakade, and Langford introduced cover trees to justify a near-linear time complexity for the neighbor search in the sizes of Q,R. Section 5.3 of Curtin's PhD (2015) pointed out that the proof of this result was wrong. The key step in the original proof attempted to show that the number of iterations can be estimated by multiplying the length of the longest root-to-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 neighbor search. The same argument was unfortunately repeated in several subsequent papers using cover trees from 2006. This paper explicitly constructs challenging datasets that provide counterexamples to the past proofs of time complexity for the cover tree construction, the k-nearest neighbor search presented at ICML 2006, and the dual-tree search algorithm published in NIPS 2009. The corrected near-linear time complexities with extra parameters are proved in another forthcoming paper by using a new compressed cover tree simplifying the original tree structure

The mergegram of a dendrogram and its stability

This paper extends the key concept of persistence within Topological Data Analysis (TDA) 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 single-linkage (SL) clustering of a finite set in a metric space. Equivalently, the 0D persistence captures only edge-lengths of a Minimum Spanning Tree (MST). Both SL dendrogram and MST are unstable under perturbations of points. We define the new stable-under-noise mergegram, which outperforms previous isometry invariants on a classification of point clouds by PersLay

A new compressed cover tree guarantees a near linear parameterized complexity for all kk-nearest neighbors search in metric spaces

This paper studies the classical problem of finding all $k$ nearest neighbors to points of a query set $Q$ in another reference set $R$ within any metric space. The well-known work by Beygelzimer, Kakade, and Langford in 2006 introduced cover trees and claimed to guarantee a near linear time complexity in the size $|R|$ of the reference set for $k=1$. Our previous work defined compressed cover trees and corrected the key arguments for $k\geq 1$ and previously unknown challenging data cases. In 2009 Ram, Lee, March, and Gray attempted to improve the time complexity by using pairs of cover trees on the query and reference sets. In 2015 Curtin with the above co-authors used extra parameters to finally prove a similar complexity for $k = 1$. Our work fills all previous gaps and substantially improves the neighbor search based on pairs of new compressed cover trees. The novel imbalance parameter of paired trees allowed us to prove a better time complexity for any number of neighbors $k\geq 1$

Structure of feeding for <i>Echinarachnius parma</i> and <i>Scaphechinus mirabilis</i> (Echinoidea, Clypeasteroida) in the Troitsa Bay, Japan Sea

Feeding of sand dollars Echinarachnius parma and Scaphechinus mirabilis (Clypeasteroida) in the Troitsa Bay, Japan Sea is investigated. Both species dwell on coarse bottom sand with the percentage of fine fraction (< 0.2 mm) no more than 3 %. Diatoms are the most important component of the sand dollars feeding, they are represented by 50 species in the ground but only 27 species in the faeces, with predominance of the cells with chloroplasts in the faeces, that indicates a selectivity of the sand dollars feeding. High similarity (0.97) of algal flora in the faeces of S. mirabilis and E. parma shows their common feeding habits. Crystals of zircon and ilmenite with specific gravity 4.7 g/cm3 are accumulated in the diverticulum of S. mirabilis though they are very rare in sandy grounds