Academic literature on the topic 'Multidimensional Hypersphere'

Multidimensional scaling of subjective color differences has shown that color stimuli are located on a hypersphere in four-dimensional space. The semantic space of color names is isomorphic with perceptual color space. A spherical four-dimensional space revealed in monkeys and fish suggests the primacy of common neuronal basis.

The study of the properties of surfaces contributes to the expansion of their use in solving various practical problems, especially if such properties can be generalized to manifolds of n-dimensional space. The most thoroughly studied are the properties of the simplest surfaces, including the properties of a sphere. That is why the simplest surfaces are most often used in practice. Each property not covered in the existing literature expands the indicated possibilities. Therefore, the purpose of this article is to identify the properties of the hypersphere unknown from the literature. Most of the properties of a circle and a sphere have been known since ancient times [1, 4, 5]. The generalized concept of a sphere into multidimensional spaces is based on the general principles of multidimensional geometry [3]. In [4], eleven basic properties of the sphere are listed and analyzed. In works [8, 10] it is shown that a circle can be considered as an isoline, and a sphere as an isosurface when modeling energy fields. In geometric modeling of energy fields with point energy sources, an essential role is played by the distances from the points of the field to the given energy sources [6, 7]. In [9], two schemes are given for determining the parameter t, taking into account the effect of the distance from the points of the field to the point sources of energy on the potentials of the points of the field. In a particular case, if this parameter is determined according to a simplified scheme with f(l)=al2, then the formula for calculating the potential of an arbitrary point of the energy field is a mathematical model of the energy field generated by the number n of point energy sources. The geometric model of the field will be a manifold that can be foliated into a one-parameter set of isospheres [8, 10]. Abstracting from the physical nature of the field, simplifying the equation for calculating the potential of an arbitrary point of the energy field and generalizing it to n-dimensional space, we can formulate the following properties: Property 1. A hypersphere can be considered as a locus of points, the sum of the squared distances from which to n given points is a constant value. Property 2. Arbitrary coefficients ki at distances li affect the parameters of the hypersphere without changing the type of surface.

Density Based Spatial Clustering of Application with Noise (DBSCAN) is one of the mostly preferred algorithm among density based clustering approaches in unsupervised machine learning, which uses epsilon neighborhood construction strategy in order to discover arbitrary shaped clusters. DBSCAN separates dense regions from low density regions and simultaneously assigns points that lie alone as outliers to unearth the hidden cluster patterns in the datasets. DBSCAN identifies dense regions by means of core point definition, detection of which are strictly dependent on input parameter definitions: ε is distance of the neighborhood or radius of hypersphere and MinPts is minimum density constraint inside ε radius hypersphere. Contrarily to classical DBSCAN’s crisp core point definition, intuitionistic fuzzy core point definition is proposed in our preliminary work to make DBSCAN algorithm capable of detecting different patterns of density by two different combinations of input parameters, particularly is a necessity for the density varying large datasets in multidimensional feature space. In this study, preliminarily proposed DBSCAN extension is studied: IFDBSCAN. The proposed extension is tested by computational experiments on several machine learning repository real-time datasets. Results show that, IFDBSCAN is superior to classical DBSCAN with respect to external & internal performance indices such as purity index, adjusted rand index, Fowlkes-Mallows score, silhouette coefficient, Calinski-Harabasz index and with respect to clustering structure results without increasing computational time so much, along with the possibility of trying two different density patterns on the same run and trying intermediary density values for the users by manipulating α margin.

Currently, smart devices of Internet of Things generate massive amount of data for different applications. However, it will expose sensitive information to external users in the process of IoT data collection, transmission, and mining. In this paper, we propose a novel indexing and searching schema based on homocentric hypersphere and similarity-aware asymmetric LSH (H2SA-ALSH) for privacy-preserved data collection and mining over IoT environments. The H2SA-ALSH collects multidimensional data objects and indexes their features according to the Euclidean norm and cosine similarity. Additionally, we design a c - k -AMIP searching algorithm based on H2SA-ALSH. Our approach can boost the performance of the maximum inner production (MIP) queries and top- k queries for a given query vector using the proposed indexing schema. Experiments show that our algorithm is excellent in accuracy and efficiency compared with other ALSH-based algorithms using real-world datasets. At the same time, our indexing scheme can protect the user’s privacy via generating similarity-based indexing vectors without exposing raw data to external users.

AbstractOne of the problems in the analysis of the set of images of a moving object is to evaluate the degree of freedom of motion and the angle of rotation. Here the intrinsic dimensionality of multidimensional data, characterizing the set of images, can be used. Usually, the image may be represented by a high-dimensional point whose dimensionality depends on the number of pixels in the image. The knowledge of the intrinsic dimensionality of a data set is very useful information in exploratory data analysis, because it is possible to reduce the dimensionality of the data without losing much information. In this paper, the maximum likelihood estimator (MLE) of the intrinsic dimensionality is explored experimentally. In contrast to the previous works, the radius of a hypersphere, which covers neighbours of the analysed points, is fixed instead of the number of the nearest neighbours in the MLE. A way of choosing the radius in this method is proposed. We explore which metric—Euclidean or geodesic—must be evaluated in the MLE algorithm in order to get the true estimate of the intrinsic dimensionality. The MLE method is examined using a number of artificial and real (images) data sets.

The main task of classical multidimensional differential geometry is to study the properties of various n-surfaces. Often these studies use torsion coefficients that are defined for any n-surface with codimension k > 1 in (n + k)-dimensional Euclidean space. For hypersurfaces, the torsion coefficients are not defined.Another important concept used to study the properties of n-surfaces is the spherical Gaussian mapping. The Gaussian mapping defined on submanifolds of Euclidean and pseudo-Euclidean spaces allows one to study the external properties of a submanifold immersed in a Euclidean or pseudo-Euclidean space. In a number of papers, the properties of the Gaussian mapping are studied, as well as the geometric characteristics of the images of submanifolds under a spherical mapping, which are submanifolds of a hypersphere or a Grassmannian.In this article, we study the local properties of the spherical image of a regular n-surface of arbitrary codimension. The spherical mapping is defined for n-surfaces with codimension greater than one in Euclidean space by means of a regular vector field. Each vector of this field at a point of the submanifold is orthogonal to the tangent space of the submanifold at the chosen point.The article uses the methods of differential and Riemannian geometry, as well as tensor analysis to study n-surfaces with a codimension greater than one. Under some additional conditions, a connection is established between the curvature tensor of a given surface and the curvature tensor of its spherical image. Under the same additional conditions, some geometric characteristics of the points of the spherical image of the original n-surface are studied.

Conventional multi-antenna transmitters use a separate RF chain (consisting of DACs, mixers, filters) and a power amplifier (PA) for each antenna element, and use modulation alphabets such as QAM/PSK for transmission. The consequences of this are, first, the RF hardware complexity, size, and cost increase with the number of antennas, and second, the linearity requirements for higher order QAM and OFDM transmissions affect the power efficiency of the amplifier in each RF chain. Load modulated array (LMA) is emerging as a promising multi-antenna transmission architecture that alleviates the aforementioned issues. An LMA uses a single central power amplifier (CPA) for the entire antenna array and no RF chains. LMA is based on the concept of load modulation, in which an antenna current proportional to the information bearing signal is achieved by modulating the antenna load impedances, while maintaining the amplifier input at a constant level. Varying the antenna load impedances with the information signal can cause a mismatch between the source impedance and the effective antenna impedance, causing power reflection into the CPA. A way to mitigate this is to ensure that the transmit signal in every channel use lies on the surface of a multidimensional hypersphere. This is called ‘phase modulation on the hypersphere (PMH)’. In this thesis, we investigate PMH signaling, detection, and precoding for LMAs in point-to-point, multiuser uplink, and multiuser downlink communication scenarios. In the first part, we consider the construction of PMH signal vectors for LMAs. Construction of PMH signal vectors is typically non-analytic (e.g., clustering, potential maximization) and hence becomes computationally and storage wise expensive. We propose random phase modulation (RPM) as an inexpensive means of constructing PMH signal vectors. The idea of RPM is extended to random phase precoding (RPP) and precoder index modulation (PIM) to devise PMH signaling schemes that achieve good performance. Indexing in time and spatial domains is also investigated. In the second part, we consider LMAs for communication in multiuser scenario. For multiuser communication on the uplink, we show using analysis and simulations that LMAs achieve superior bit error performance compared to other single RF chain multi-antenna transmission architectures in the literature. To exploit this performance advantage in large systems, we propose low complexity multiuser signal detection algorithms based on message passing and Monte Carlo sampling techniques. For multiuser communication on the downlink, we design a block diagonalizing precoder that nulls multiuser interference at the user terminals while ensuring that the precoded signal vector lies on the hypersphere. Further, when the antenna load impedances are tuned using discrete values from a finite set, the support of the precoded signal vector is finite. For this setting, we propose an iterative precoding algorithm using the generalized least square error (GLSE) framework. In the final part, we propose a hybrid signaling scheme using LMAs and channel modulation (CM), wherein additional information bits are conveyed through the ON/OFF status of parasitics placed near the antenna array. For this scheme, we exploit the inherent sparsity in the signal vectors for multiuser detection on the uplink. We also incorporate CM in the GLSE framework for multiuser precoding on the downlink.