Points In Euclidean Space Research Articles

Approximate Distance Oracle for Fault-Tolerant Geometric Spanners

In this paper, we present approximate distance and shortest-path oracles for fault-tolerant Euclidean spanners motivated by the routing problem in real-world road networks. A fault-tolerant Euclidean spanner for a set of points in Euclidean space is a graph in which, despite the deletion of small number of any points, the distance between any two points in the damaged graph is an approximation of their Euclidean distance. Given a fault-tolerant Euclidean spanner and a small approximation factor, our data structure allows us to compute an approximate distance between two points in the damaged spanner in constant time when a query involves any two points and a small set of failed points. Additionally, by incorporating additional data structures, we can return a path itself in time almost linear in the length of the returned path. Both data structures require near-linear space.

The Four Point Condition: An Elementary Tropicalization of Ptolemy’s Inequality

Ptolemy’s inequality is a classic relationship between the distances among four points in Euclidean space. Another relationship between six distances is the 4-point condition, an inequality satisfied by the lengths of the six paths that join any four points of a metric (or weighted) tree. The 4-point condition also characterizes when a finite metric space can be embedded in such a tree. The curious observer might realize that these inequalities have similar forms: if one replaces addition and multiplication in Ptolemy’s inequality with maximum and addition, respectively, one obtains the 4-point condition. We show that this similarity is more than a coincidence. We identify a family of Ptolemaic inequalities in CAT-spaces parametrized by a real number and show that a certain limit involving these inequalities, as the parameter goes to negative infinity, yields the 4-point condition, giving an elementary proof that the latter is the tropicalization of Ptolemy’s inequality.

Boosting 3D Object Detection with Density-Aware Semantics-Augmented Set Abstraction.

In recent years, point cloud-based 3D object detection has seen tremendous success. Previous point-based methods use Set Abstraction (SA) to sample the key points and abstract their features, which did not fully take density variation into consideration in point sampling and feature extraction. The SA module can be split into three parts: point sampling, grouping and feature extraction. Previous sampling methods focus more on distances among points in Euclidean space or feature space, ignoring the point density, thus making it more likely to sample points in Ground Truth (GT) containing dense points. Furthermore, the feature extraction module takes the relative coordinates and point features as input, while raw point coordinates can represent more informative attributes, i.e., point density and direction angle. So, this paper proposes Density-aware Semantics-Augmented Set Abstraction (DSASA) for solving the above two issues, which takes a deep look at the point density in the sampling process and enhances point features using onefold raw point coordinates. We conduct the experiments on the KITTI dataset and verify the superiority of DSASA.

Normal approximation via non-linear exchangeable pairs

We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract bounds on the normal and Gamma approximation of certain functionals in the Wasserstein distance. Moreover, we illustrate the relevance of this approach by means of three instances of situations to which it can be applied: Functionals of independent random variables, finite population statistics and functionals on finite groups. In the independent case, and in particular for symmetric $U$-statistics, we demonstrate in which respect this approach yields fundamentally better bounds than those in the existing literature. Finally, we apply our results to provide Wasserstein bounds in a CLT for subgraph counts in geometric random graphs based on $n$ i.i.d. points in Euclidean space as well as to the normal approximation of Pearson's statistic.

1-mean and 1-medoid 2-clustering problem with arbitrary cluster sizes: Complexity and approximation

We consider the following 2-clustering problem. Given N points in Euclidean space, partition it into two subsets (clusters) so that the sum of squared distances between the elements of the clusters and their centers would be minimum. The center of the first cluster coincides with its centroid (mean) while the center of the second cluster should be chosen from the set of the initial points (medoid). It is known that this problem is NP-hard if the cardinalities of the clusters are given as a part of the input. In this paper we prove that the problem remains NP-hard in the case of arbitrary clusters sizes and suggest a 2-approximation polynomial-time algorithm for this problem.

Configuration spaces, multijet transversality, and the square-peg problem

We prove a transversality “lifting property” for compactified configuration spaces as an application of the multijet transversality theorem: given a submanifold of configurations of points on an embedding of a compact manifold M in Euclidean space, we can find a dense set of smooth embeddings of M for which the corresponding configuration space of points is transverse to any submanifold of the configuration space of points in Euclidean space, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide an attractive proof of the square-peg problem: there is a dense family of smoothly embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a dense family of smoothly embedded circles in Rn where each simple closed curve has an odd number of inscribed square-like quadrilaterals.

On prescribing total orders and preorders to pairwise distances of points in Euclidean space

We show that any total preorder on a set of the form (X2) where X has n elements coincides with the order on pairwise distances of some point collection of size n in Rn−1. For total orders, a collection of n points in Rn−2 suffices. These bounds turn out to be optimal. We also find an optimal bound in a bipartite version for total preorders and a near-optimal bound for a bipartite version for total orders. Our arguments include tools from convexity and positive semidefinite quadratic forms.

T-Maps-Based Tolerance Analysis of Composites Assembly Involving Compensation Strategies

Abstract Clamping force and shimming are two important compensation processes in the composite assembly. Their effects on variation propagation should be investigated in tolerance analysis. The paper presents a tolerance analysis method for composites assembly based on the T-Maps method, mainly concerning the anisotropic variations accumulation and propagation where there is the clamping force modification and the shimming. Variations of the composite parts in different directions are represented by the T-Maps. Since the different axial deviations are represented in the same Euclidean point-space, the T-Maps-based tolerance analysis of the composite parts assembly provides more accurate and reliable results. Compensation processes, the clamping force, and the shimming, on assembly tolerance synthesis of the composite parts, are analyzed clearly in the T-Map. This procedure is found to be effective for the anisotropy-oriented assembly tolerance analysis, especially concerning about effect of the clamping force and the shimming on variations accumulation and propagation. The assembly of an aircraft composite elevator is considered to demonstrate the effectiveness of the T-Maps-based method. The procedures outlined in the paper are quite general and can be used for assembly tolerance analysis of anisotropic parts.

Rates of Convergence for the Continuum Limit of Nondominated Sorting

Nondominated sorting is a discrete process that sorts points in Euclidean space according to the coordinatewise partial order, and is used to rank feasible solutions to multiobjective optimization problems. It was previously shown that nondominated sorting of random points has a Hamilton-Jacobi equation continuum limit. We prove quantitative error estimates for the convergence of nondominated sorting to its continuum limit Hamilton-Jacobi equation. Our proof uses the maximum principle and viscosity solution machinery, along with new semiconvexity estimates for domains with corner singularities.

Effectiveness of Deep Learning Trained on SynthCity Data for Urban Point-Cloud Classification

3D object recognition is one of the most popular areas of study in computer vision. Many of the more recent algorithms focus on indoor point clouds, classifying 3D geometric objects, and segmenting outdoor 3D scenes. One of the challenges of the classification pipeline is finding adequate and accurate training data. Hence, this article seeks to evaluate the accuracy of a synthetically generated data set called SynthCity, tested on two mobile laser-scan data sets. Varying levels of noise were applied to the training data to reflect varying levels of noise in different scanners. The chosen deep-learning algorithm was Kernel Point Convolution, a convolutional neural network that uses kernel points in Euclidean space for convolution weights.

Вычислительная экспансия в графы ближайших соседей: статистики и размерности пространства

The distributions of graphs of the first nearest neighbors by the number of disconnected fragments, fragments by the number of vertices, and vertices by the degrees of incoming edges, depending on the number of vertices of the graph, are investigated. Two situations are considered: when the matrix of random distances is given directly, and when random coordinates of points in Euclidean space of a given dimension are given. In the course of a computational experiment, it is shown that with an increase in the dimension of the space, the statistics of the first and second variants converge. For dimensions above the fifth, the degree distributions of the vertices could be used approximately at the same significance level.

Configurations in fractals

We define the manifold of configurations to be the quotient set of k points in Euclidean space identified under congruence, and prove that compact subsets of ℝd, d≥2, of large Hausdorff dimension have a non-null set of configurations in them. Our method simplifies previous work and achieves a better dimensional threshold.

Band contour-extraction method based on conformal geometrical algebra for space tumbling targets.

To remove space debris actively, the key step is choosing a suitable capturing method. The size characteristics of space debris are an important factor that affects the capturing method. This paper presents a contour-extracting algorithm for rolling targets, such as space debris. The representation of the point in Euclidean space is converted to a conformal space to obtain candidate contour points through the computational geometric and topological relationships among the points, lines, and surfaces in a high-dimensional space. By combining this conversion with the multiview stitching algorithm, the complete candidate contour point model is obtained. The final contour points are extracted by determining the concave and convex points in conformal space. Simulation results prove that the unified expression of the geometric features in conformal space not only reduces the computational complexity but also increases the speed of the contour point extraction. The method proposed in this paper is superior to other methods in terms of measurement accuracy, speed, and robustness; in particular, the accuracy increased by 60%.

On the classification of framed rectifying curves in Euclidean space

There are many studies on regular rectifying curves in classical differential geometry, and important results have been obtained. However, these studies are limited for a smooth curve with singular points. To examine such curves and surfaces, the concept of framed curve, which is the general form of regular and Legendre curves, is used. Framed curves are defined as curves that have a moving frame with singular points in Euclidean space. We investigate framed rectifying curves via the dilation of framed curves on S2 in . Moreover, the result of dilation of framed curves is the framed rectifying curve or not. We give some classifications for the dilation of framed curves. Finally, we give some related examples with their figures.

Dk Gravitational Instantons as Superpositions of Atiyah–Hitchin and Taub–NUT Geometries

Abstract We obtain Dk ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as nonlinear superpositions of the moduli space of centred SU(2) monopoles, equipped with the Atiyah–Hitchin metric, and k copies of the Taub–NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we show that the constituent Atiyah–Hitchin and Taub–NUT geometries arise as boundary components of the five-dimensional geometry as the adiabatic parameter is taken to zero.

A Geometry Feature Aggregation Method for Point Cloud Classification and Segmentation

In spite of the good performance of convolutional neural network (CNN) and graph neural network (GNN) in 3D point cloud classification and segmentation at present, to aggregate local information of point clouds and improve the robustness of geometric transformation are still challenging problems. In order to tackle the problems, we propose Geometry Feature Aggregation Network (GFA-Net), which can effectively learn the context information of each point to aggregate local information, so as to enhance the robustness of rotation and translation. Compared with the current popular method GNN that convolves on nearby points in Euclidean space, GFA-Net can better aggregate the geometric features around the points. GFA-Net uses the Laplacian feature mapping to reduce dimensions, and aggregates the nearest neighbor features in the space after dimensionality reduction, and fuses them with the nearest neighbor features of Euclidean space, so as to better obtain the geometric features of each point. Then, points are grouped with geometric features, so that nearby points are insensitive to geometric transformations such as rotation and translation. This method allows GFA-Net to better obtain holistic geometry features, such as symmetry. In addition, we use attention mechanism instead of pooling, so that important neighborhood information can be learned automatically and information loss can be reduced. We conduct extensive experiments on public datasets ModelNet40 and ShapeNet Part. The experimental results show that GFA-Net achieves very good performance, which is very close to the current state-of-the-art methods, and GFA-Net has better robustness.

МЕТОДИКА СИНТЕЗА ПОДСИСТЕМЫ ИНТЕЛЛЕКТУАЛЬНОГО МОНИТОРИНГА ИНФОРМАЦИОННО}ТЕЛЕКОММУНИКАЦИОННОЙ СЕТИ СИТУАЦИОННОГО ЦЕНТРА

Introduction: in accordance with the task of forming a system of distributed situational centers of public authorities, it is necessary to interface heterogeneous segments of the department's information and telecommunications networks into a geographically distributed infrastructure with the construction of a subsystem for monitoring the state of its elements on it. The purpose of the study: based on the use of basic concepts of structural synthesis methods, structural analysis of network infrastructures, as well as parametric synthesis procedures, to develop a methodology for the synthesis of a new generation monitoring subsystem based on intelligent methods for identifying the state of the network. Results: the structuring of the controlled space by the terms "monitoring zone"," critical element"," state classes "form the basis of new methods of intelligent monitoring that are" insensitive " to the properties of the constant evolution of network infrastructures. The proposed method of synthesis of the intelligent monitoring subsystem consists of sequentially performed stages of structural synthesis, parametric synthesis and structural analysis of the network. Practical significance: the successive stages of the methodology using graph distance measurement procedures and the modified k-means algorithm allow not only to identify the type of network state, but also to reasonably, using instrumental calculation methods, present in the interests of the decision support system sets of acceptable values of the main parameters and probabilistic-temporal characteristics of the monitoring subsystem for subsequent reconfiguration of the network and preventing its transition to an inoperable state. Discussion: the modification of the k-means algorithm proposed in the study differs in that in the classical algorithm, work is carried out on points of Euclidean space, and in the proposed method we are talking about a graph space with metrics in the form of graph distances, while data clouds are used as initial data for classification in the k-means algorithm as disordered data sets that are not tied to any of the measurement scales, and in the proposed algorithm, the data cloud is represented by a set of graphs in a given topological space of graph metrics, describing the state of the network in time.

Combinatorial and Computational Investigations of Neighbor-Joining Bias.

The Neighbor-Joining algorithm is a popular distance-based phylogenetic method that computes a tree metric from a dissimilarity map arising from biological data. Realizing dissimilarity maps as points in Euclidean space, the algorithm partitions the input space into polyhedral regions indexed by the combinatorial type of the trees returned. A full combinatorial description of these regions has not been found yet; different sequences of Neighbor-Joining agglomeration events can produce the same combinatorial tree, therefore associating multiple geometric regions to the same algorithmic output. We resolve this confusion by defining agglomeration orders on trees, leading to a bijection between distinct regions of the output space and weighted Motzkin paths. As a result, we give a formula for the number of polyhedral regions depending only on the number of taxa. We conclude with a computational comparison between these polyhedral regions, to unveil biases introduced in any implementation of the algorithm.

On the Complexity of SomeMax–Min Clustering Problems

Two similar problems of searching for a family of disjoint subsets (clusters) in a finite set of points in Euclidean space are considered. In these problems, the size of the smallest cluster should be maximized so that in each cluster the intracluster quadratic variation of the points with respect to the center of the cluster would not exceed a given (constant) fraction of the total quadratic variation of the points of the input set with respect to its centroid. In the first problem, the centers of intracluster variations are arbitrary points of the space given at the input. In the second problem, the centers of the intracluster variation are unknown (to be found) but must lie in the input set. Both problems are proved to be NP-hard even on the real line both in the general case when the number of the clusters is a part of the input and in the parametric case when the number of the clusters is fixed.

Finding the largest triangle in a graph in expected quadratic time

Finding the largest triangle in an n-nodes edge-weighted graph belongs to a set of problems all equivalent under subcubic reductions. Namely, a truly subcubic algorithm for any one of them would imply that they are all subcubic. A recent strong conjecture states that none of them can be solved in less than Θ(n3) time, but this negative result does not rule out the possibility of algorithms with average, rather than worst-case, subcubic running time. Indeed, in this work we describe the first truly-subcubic average complexity procedure for this problem for graphs whose edge lengths are uniformly distributed in [0,1]. Our procedure finds the largest triangle in average quadratic time, which is the best possible complexity of any algorithm for this problem. We also give empirical evidence that the quadratic average complexity holds for many other random distributions of the edge lengths. A notable exception is when the lengths are distances between random points in Euclidean space, for which the algorithm takes average cubic time.