Point (geometry)

About: Point (geometry) is a research topic. Over the lifetime, 16807 publications have been published within this topic receiving 186822 citations. The topic is also known as: fixed point (survey).

Curvilinear component analysis: a self-organizing neural network for nonlinear mapping of data sets

Abstract: We present a new strategy called "curvilinear component analysis" (CCA) for dimensionality reduction and representation of multidimensional data sets. The principle of CCA is a self-organized neural network performing two tasks: vector quantization (VQ) of the submanifold in the data set (input space); and nonlinear projection (P) of these quantizing vectors toward an output space, providing a revealing unfolding of the submanifold. After learning, the network has the ability to continuously map any new point from one space into another: forward mapping of new points in the input space, or backward mapping of an arbitrary position in the output space.

Angle-based outlier detection in high-dimensional data

Abstract: Detecting outliers in a large set of data objects is a major data mining task aiming at finding different mechanisms responsible for different groups of objects in a data set All existing approaches, however, are based on an assessment of distances (sometimes indirectly by assuming certain distributions) in the full-dimensional Euclidean data space In high-dimensional data, these approaches are bound to deteriorate due to the notorious "curse of dimensionality" In this paper, we propose a novel approach named ABOD (Angle-Based Outlier Detection) and some variants assessing the variance in the angles between the difference vectors of a point to the other points This way, the effects of the "curse of dimensionality" are alleviated compared to purely distance-based approaches A main advantage of our new approach is that our method does not rely on any parameter selection influencing the quality of the achieved ranking In a thorough experimental evaluation, we compare ABOD to the well-established distance-based method LOF for various artificial and a real world data set and show ABOD to perform especially well on high-dimensional data

On the Complexity of Some Common Geometric Location Problems

Abstract: Given n demand points in the plane, the p-center problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demand point to its respective nearest supply point. The p-median problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the p-center and the p-median problems relative to both the Euclidean and the rectilinear metrics are NP-hard. In fact, we prove that it is NP-hard even to approximate the p-center problems sufficiently closely. The reductions are from 3-satisfiability.

Nonparametric methods in change-point problems

Abstract: Preface. Introduction: Goals and problems of change point detection. 1. Preliminary considerations. 2. State-of-the-art review. 3. A posteriori change point problems. 4. Sequential change point detection problems. 5. Disorder detection of random fields. 6. Applications of nonparametric change point detection methods. 7. Proofs, new results and technical details. References.

Computing Dirichlet Tessellations in the Plane

Abstract: A finite set of distinct points divides the plane into polygonal regions, each region containing one of the points and comprising that part of the plane nearer to its defining point than to any other. The resultant planar subdivision is called the Dirichlet tessellation; it is one of the most useful constructs associated with such a point configuration. The regions, which we call tiles, are also known as Voronoi or Thiessen polygons. We describe a recursive algorithm for computing the tessellation in a highly efficient way, and discuss the problems which arise in its implementation. Samples of graphical output demonstrate the application of the program on a modest scale; its efficiency allows its application to large sets of data, and detailed discussion of space and time considerations is given, based in part on theoretical predictions and in part on test runs on up to 10,000 points.