Points In Euclidean Space Research Articles

An Approximation Algorithm for a Problem of Partitioning a Sequence into Clusters with Constraints on Their Cardinalities

We consider the problem of partitioning a finite sequence of points in Euclidean space into a given number of clusters (subsequences) minimizing the sum over all clusters of intracluster sums of squared distances of elements of the clusters to their centers. It is assumed that the center of one of the desired clusters is the origin, while the centers of the other clusters are unknown and are defined as the mean values of cluster elements. Additionally, there are a few structural constraints on the elements of the sequence that enter the clusters with unknown centers: (1) the concatenation of indices of elements of these clusters is an increasing sequence, (2) the difference between two consequent indices is lower and upper bounded by prescribed constants, and (3) the total number of elements in these clusters is given as an input. It is shown that the problem is strongly NP-hard. A 2-approximation algorithm that is polynomial for a fixed number of clusters is proposed for this problem.

Representation of a Smooth Isometric Deformation of a Planar Material Region into a Curved Surface

We consider the problem of characterizing the smooth, isometric deformations of a planar material region identified with an open, connected subset {mathcal{D}} of two-dimensional Euclidean point space mathbb{E}^{2} into a surface {mathcal{S}} in three-dimensional Euclidean point space mathbb{E}^{3}. To be isometric, such a deformation must preserve the length of every possible arc of material points on {mathcal{D}}. Characterizing the curves of zero principal curvature of {mathcal{S}} is of major importance. After establishing this characterization, we introduce a special curvilinear coordinate system in mathbb{E}^{2}, based upon an à priori chosen pre-image form of the curves of zero principal curvature in {mathcal{D}}, and use that coordinate system to construct the most general isometric deformation of {mathcal{D}} to a smooth surface {mathcal{S}}. A necessary and sufficient condition for the deformation to be isometric is noted and alternative representations are given. Expressions for the curvature tensor and potentially nonvanishing principal curvature of {mathcal{S}} are derived. A general cylindrical deformation is developed and two examples of circular cylindrical and spiral cylindrical form are constructed. A strategy for determining any smooth isometric deformation is outlined and that strategy is employed to determine the general isometric deformation of a rectangular material strip to a ribbon on a conical surface. Finally, it is shown that the representation established here is equivalent to an alternative previously established by Chen, Fosdick and Fried (J. Elast. 119:335–350, 2015).

An Information and Control Framework for Optimizing User-Compliant Human–Computer Interfaces

We consider a general framework for a human–computer interface whereby the human's knowledge is represented as a point in Euclidean space, the intention of the human is signaled to the computer over a noisy channel, and the computer queries the human in a manner that is amenable to human operation. With these constraints at hand, we demonstrate a class of systems that are nonetheless information-theoretically optimal in that the computer very rapidly hones in on the intent of the human. Much recent work on feedback information theory has been dedicated to the exploration of methods by which optimal feedback may be derived for the purpose of expediting the communication of a message point between an inanimate encoder and decoder. Our framework not only takes advantage of previous work to demonstrate its communication optimality from this perspective as well as from an information-theoretic perspective but also contributes two distinct advantages. First, our framework provides a simplified method based on optimal transport theory to generate optimal feedback signals between the computer and human in high dimension, while still preserving communication optimality. Second, our framework specifically lends itself to the integration of a human user by attempting to moderate the difficulty of the task presented to the user, while still preserving optimality. We demonstrate applications of our framework within the context of multi-agent brain-computer interfaces.

Towards Topological Mechanisms Underlying Experience Acquisition and Transmission in the Human Brain.

Experience is a process of awareness and mastery of facts or events, gained through actual observation or second-hand knowledge. Recent findings reinforce the idea that a naturalized epistemological approach is needed to further advance our understanding of the nervous mechanisms underlying experience. This essay is an effort to build a coherent topological-based framework able to elucidate particular aspects of experience, e.g., how it is acquired by a single individual, transmitted to others and collectively stored in form of general ideas. Taking into account concepts from neuroscience, algebraic topology and Richard Avenarius' philosophical analytical approach, we provide a scheme which is cast in an empirically testable fashion. In particular, we emphasize the foremost role of variants of the Borsuk-Ulam theorem, which tells us that, when a pair of opposite (antipodal) points on a sphere are mapped onto a single point in Euclidean space, the projection provides a description of both antipodal points. These antipodes stand for nervous functions and activities of the brain correlated with the mechanisms of acquisition and transmission of experience.

Separating subadditive euclidean functionals

AbstractThe classical Beardwood‐Halton‐Hammersly theorem (1959) asserts the existence of an asymptotic formula of the form for the minimum length of a Traveling Salesperson Tour throuh n random points in the unit square, and in the decades since it was proved, the existence of such formulas has been shown for other such Euclidean functionals on random points in the unit square as well. Despite more than 50 years of attention, however, it remained unknown whether the minimum length TSP through n random points in was asymptotically distinct from its natural lower bounds, such as the minimum length spanning tree, the minimum length 2‐factor, or, as raised by Goemans and Bertsimas, from its linear programming relaxation. We prove that the TSP on random points in Euclidean space is indeed asymptotically distinct from these and other natural lower bounds, and show that this separation implies that branch‐and‐bound algorithms based on these natural lower bounds must take nearly exponential () time to solve the TSP to optimality, even in average case. This is the first average‐case superpolynomial lower bound for these branch‐and‐bound algorithms (a lower bound as strong as was not even been known in worst‐case analysis). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 375–403, 2017

Polynomial-time approximation scheme for a problem of partitioning a finite set into two clusters

We consider the strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters of given cardinalities under the minimum criterion for the sum over the clusters of the intracluster sums of squared distances from elements of the cluster to its center. It is assumed that the center of one of the clusters is given (without loss of generality, at the origin). The center of the second cluster is unknown and is defined as the mean value over all elements in this cluster. A polynomial-time approximation scheme (PTAS) is provided.

Improper Poisson line process as SIRSN in any dimension

Aldous has introduced a notion of scale-invariant random spatial network (SIRSN) as a mathematical formalization of road networks. Intuitively, those are random processes that assign a route between each pair of points in Euclidean space, while being invariant under rotation, translation, and change of scale, and such that the routes are not too long and mainly on ``main roads''. The only known example was somewhat artificial since invariance had to be added at the end of the construction. We prove that the network of geodesics in the random metric space generated by a Poisson line process marked by speeds according to a power law is a SIRSN, in any dimension. Along the way, we establish bounds comparing Euclidean balls and balls for the random metric space. We also prove that in dimension more than two, the geodesics have ``many directions'' near each point where they are not straight.

Multiresolution on spherical curves

In this paper, we present an approximating multiresolution framework of arbitrary degree for curves on the surface of a sphere. Multiresolution by subdivision and reverse subdivision allows one to decrease and restore the resolution of a curve, and is typically defined by affine combinations of points in Euclidean space. While translating such combinations to spherical space is possible, ensuring perfect reconstruction of the curve remains challenging. Hence, current spherical multiresolution schemes tend to be interpolating or midpoint-interpolating, as achieving perfect reconstruction in these cases is more straightforward. We use a simple geometric construction for a non-interpolating and non-midpoint-interpolating multiresolution scheme on the sphere, which is made up of easily generalized components and based on a modified Lane–Riesenfeld algorithm.

A direct verification argument for the Hamilton–Jacobi equation continuum limit of nondominated sorting

Nondominated sorting is a combinatorial algorithm that sorts points in Euclidean space into layers according to a partial order. It was recently shown that nondominated sorting of random points has a Hamilton–Jacobi equation continuum limit. The original proof, given in Calder et al. (2014), relies on a continuum variational problem. In this paper, we give a new proof using a direct verification argument that completely avoids the variational interpretation. We believe this may be generalized to apply to other stochastic homogenization problems for which there is no obvious underlying variational principle.

A fully polynomial-time approximation scheme for a sequence 2-cluster partitioning problem

We consider a strongly NP-hard problem of partitioning a finite sequence of points in Euclidean space into the two clustersminimizing the sum over both clusters of intra-cluster sums of squared distances from the clusters elements to their centers. The sizes of the clusters are fixed. The centroid of the first cluster is defined as the mean value of all vectors in the cluster, and the center of the second cluster is given in advance and equals 0. Additionally, the partition must satisfy the restriction that for all vectors in the first cluster the difference between the indices of two consequent points from this cluster is bounded from below and above by some given constants.We present a fully polynomial-time approximation scheme for the case of fixed space dimension.

Focal point computation and homogeneous geometrical transformation for linear curves

Summary In computer graphics, computer aided design, solid modelling and robotics and pattern recognition evaluation of focal point for a collection of line segments is the bottleneck problem. In the computer era the significant importance is given to this problem in present research world, however resolving its complexity has persisted imperceptible. This paper presents an efficient and optimal algorithm to evaluate intersection point in Euclidean space by evolving the mathematical analysis. Analysis of mathematics of the algorithm is rather seems to be intricate but implementation of algorithm is simple and straightforward.

Fully polynomial-time approximation scheme for a special case of a quadratic Euclidean 2-clustering problem

The strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters of given sizes (cardinalities) minimizing the sum (over both clusters) of the intracluster sums of squared distances from the elements of the clusters to their centers is considered. It is assumed that the center of one of the sought clusters is specified at the desired (arbitrary) point of space (without loss of generality, at the origin), while the center of the other one is unknown and determined as the mean value over all elements of this cluster. It is shown that unless P = NP, there is no fully polynomial-time approximation scheme for this problem, and such a scheme is substantiated in the case of a fixed space dimension.

Combining and congruence evaluation of phylogenetic signals from different genes based on geometric approach

A new Euclidean distance based algorithm is used for analysis of congruence and combining molecular genetic data. This approach is called geometric, since Euclidean distance satisfies all metric axioms and the points representing the sequences can be placed in a geometric space without distorting the mutual distances and can be endowed with the coordinates in this space. Geometricness of Euclidean distances allows to apply to molecular data methods of multivariate analysis, which are relevant for intra- and interspecies variability investigating, visualization of possible directions of evolution, combining data and evaluation of the congruence of phylogenetic signals. The algorithm is used for the analysis of more than 1500 nucleotide sequences of two nuclear (apoB, brca1) and two mitochondrial (co1, cytb) genes of 15 Palaearctic and Nearctic shrews species of genus Sorex (Soricidae, Eulipotyphla). All sequences of each gene are represented as a set of points in Euclidean space. Centroids of a set of points belonging to the same species are calculated. The matrix of Euclidean distances between the species centroids is calculated for each gene. Mantel test is applied to estimate pairwise similarity (congruence) of interspecies distances matrices relating to different genes. nDNA genes congruence is equal 0.961, mtDNA – 0.748. All matrices of the interspecies distances are combined into a joint matrix by weighing. Joint genetic space for all species is built by principal coordinate method from the joint matrix. Several variability directions reflecting evolutionary events of different scale are visualized in a joint genetic space. In addition, the joint matrix of interspecies distances is used for building a phylogenetic tree which is consistent with the zoological systematics accepted for today. This confirms the efficiency of our proposed method.

Adjusted least squares fitting of algebraic hypersurfaces

We consider the problem of fitting a set of points in Euclidean space by an algebraic hypersurface. We assume that points on a true hypersurface, described by a polynomial equation, are corrupted by zero mean independent Gaussian noise, and we estimate the coefficients of the true polynomial equation. The adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator. The adjusted least squares estimator is based on constructing a quasi-Hankel matrix, which is a bias-corrected matrix of moments. For the case of unknown noise variance, the estimator is defined as a solution of a polynomial eigenvalue problem. In this paper, we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for an arbitrary set of monomials in the polynomial equation.

NP-hardness of some Quadratic Euclidean 2-clustering problems

Some problems of partitioning a finite set of points of Euclidean space into two clusters are considered. In these problems, the following criteria are minimized: (1) the sum over both clusters of the sums of squared pairwise distances between the elements of the cluster and (2) the sum of the (multiplied by the cardinalities of the clusters) sums of squared distances from the elements of the cluster to its geometric center. Additionally, another problem close to (2) is considered, in which the desired center of one of the clusters is given as input, while the center of the other cluster is unknown (is the variable to be optimized) as in problem (2). Two variants of the problems are analyzed, in which the cardinalities of the clusters are parts of the input or optimization variables. It is proved that all the considered problems are strongly NP-hard and that, in general, there is no fully polynomial-time approximation scheme for them (unless P = NP).

Bounding Interference in Wireless Ad Hoc Networks With Nodes in Random Position

Given a set of positions for wireless nodes, the interference minimization problem is to assign a transmission radius (i.e., a power level) to each node such that the resulting communication graph is connected while minimizing the maximum (respectively, average) interference. We consider the model introduced by von Rickenbach (2005), in which each wireless node is represented by a point in Euclidean space on which is centered a transmission range represented by a ball, and edges in the corresponding graph are symmetric. The problem is NP-complete in two or more dimensions (Buchin 2008), and no polynomial-time approximation algorithm is known. We show how to solve the problem efficiently in settings typical for wireless ad hoc networks. If nodes are represented by a set P of n points selected uniformly and independently at random over a d-dimensional rectangular region, then the topology given by the closure of the Euclidean minimum spanning tree of P has O(log n) maximum interference with high probability and O(1) expected interference. We extend the first bound to a general class of communication graphs over a broad set of probability distributions. We present a local algorithm that constructs a graph from this class; this is the first local algorithm to provide an upper bound on expected maximum interference. Finally, we disprove a conjecture of Devroye and Morin (2012) relating the maximum interference of the Euclidean minimum spanning tree to the optimal maximum interference attainable.

Generalized Weiszfeld Algorithms for Lq Optimization.

In many computer vision applications, a desired model of some type is computed by minimizing a cost function based on several measurements. Typically, one may compute the model that minimizes the L2 cost, that is the sum of squares of measurement errors with respect to the model. However, the Lq solution which minimizes the sum of the qth power of errors usually gives more robust results in the presence of outliers for some values of q, for example, q = 1. The Weiszfeld algorithm is a classic algorithm for finding the geometric L1 mean of a set of points in Euclidean space. It is provably optimal and requires neither differentiation, nor line search. The Weiszfeld algorithm has also been generalized to find the L1 mean of a set of points on a Riemannian manifold of non-negative curvature. This paper shows that the Weiszfeld approach may be extended to a wide variety of problems to find an Lq mean for 1 ≤ q <; 2, while maintaining simplicity and provable convergence. We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global Lq optimum) and multiple rotation averaging (for which no such proof exists). Experimental results of Lq optimization for rotations show the improved reliability and robustness compared to L2 optimization.

Representation for a Smooth Isometric Mapping from a Connected Planar Domain to a Surface

A representation theorem for a smooth isometric mapping of a flat, connected domain \(\mathcal {D}\) in two-dimensional Euclidean point space \(\mathcal {E}^{2}\) into a surface \(\mathcal {S}\) in three-dimensional Euclidean point space \(\mathcal {E}^{3}\) is presented. The form of the mapping is shown to be necessary and sufficient to describe any such smooth isometry. Importantly, this work is not based upon the hypothesis that the mapped surface is ruled. In general, a mapping from a flat planar domain into a ruled surface is far from being isometric, and the property of being ruled is a partial consequence of our representation theorem.

Minsum Location Extended to Gauges and to Convex Sets

One of the oldest and richest problems from continuous location science is the famous Fermat-Torricelli problem, asking for the unique point in Euclidean space that has minimal distance sum to n given (non-collinear) points. Many natural and interesting generalizations of this problem were investigated, e.g., by extending it to non-Euclidean spaces and modifying the used distance functions, or by generalizing the configuration of participating geometric objects. In the present paper, we extend the Fermat-Torricelli problem in a two-fold way: more general than for normed spaces, the unit balls of our spaces are compact convex sets having the origin as interior point (but without symmetry condition), and the n given objects can be general convex sets (instead of points). We combine these two viewpoints, and the presented sequence of new theorems follows in a comparing sense that of theorems known for normed spaces. Some of these results holding for normed spaces carry over to our more general setting, and others not. In addition, we present analogous results for related questions, like, e.g., for Heron's problem. And finally we derive a collection of results holding particularly for the Euclidean norm.

On the Conjectures Regarding the 4-Point Atiyah Determinant

For the case of 4 points in Euclidean space, we present a computer aided proof of Conjectures II and III made by Atiyah and Sutcliffe regarding Atiyah's determinant along with an elegant factorization of the square of the imaginary part of Atiyah's determinant.