A Word Selection Method for Producing Interpretable Distributional Semantic Word Vectors

Distributional semantic models represent the meaning of words as vectors. We introduce a selection method to learn a vector space that each of its dimensions is a natural word. The selection method starts from the most frequent words and selects a subset, which has the best performance. The method produces a vector space that each of its dimensions is a word. This is the main advantage of the method compared to fusion methods such as NMF, and neural embedding models. We apply the method to the ukWaC corpus and train a vector space of N=1500 basis words. We report tests results on word similarity tasks for MEN, RG-65, SimLex-999, and WordSim353 gold datasets. Also, results show that reducing the number of basis vectors from 5000 to 1500 reduces accuracy by about 1.5-2%. So, we achieve good interpretability without a large penalty. Interpretability evaluation results indicate that the word vectors obtained by the proposed method using N=1500 are more interpretable than word embedding models, and the baseline method. We report the top 15 words of 1500 selected basis words in this paper.

The affine connection

The connection and the covariant derivative are treated. Connection coefficients are introduced in their role of expressing the change in the coordinate basis vectors between neighbouring points. The covariant derivative of a vector is then defined. Next we relate the connection to the metric, and obtain the Levi-Civita connection. The logic concerning what is defined and what is derived is explained carefuly. The notion of a derivative along a curve is defined. The emphasis through is on clarity and avoiding confusions arising from the plethora of concepts and symbols.

Hybrid basis vector based underdetermined beamforming algorithm in optimized antenna reconfiguration

Optimized positioning of antenna to obtain the best beam forming solution is adopted in this research. Non-uniform linear array-based beamforming algorithms have the challenge of placing the array of antennas in positions that would implement best beamforming outputs. This paper attempts to obtain the optimized beam forming by tuning the sparse Bayesian learning based algorithm. The parameters used for tuning involve choosing the hybrid basis vector for creating the steering vector while at the same time developing the optimized position of the antennas. Basis vectors are the building blocks of the steering vector developed for the beamforming algorithm that finds the angle of arrival in antennas. Reconfiguration of antennas is carried out using particle swarm optimization (PSO) algorithm and the basis vectors are generated using two different ways. One by cumulating similar basis vectors and another by cumulating two different basis vectors. The performance of accurate detection of angle of arrival in the beamforming algorithm is analyzed and results are discussed. This basis vector and antenna distance optimization is adopted on the sparse Bayesian learning paradigm. Performance evaluation of these optimizations in the algorithm is realised by validating the mean square error (MSE) versus signal to noise ratio (SNR) graphs for both the cumulative basis vector and hybrid basis vector cases.

Surfaces and Curves Induced by Nonlinear Schrödinger-Type Equations and Their Spin Systems

In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significantly simplify the procedure for establishing equivalence between nonlinear integrable equations from different areas of physics, which in turn open up opportunities to easily find their solutions. In this paper, we study the symmetry between differential geometry of surfaces/curves and some integrable generalized spin systems. In particular, we investigate the gauge and geometrical equivalence between the local/nonlocal nonlinear Schrödinger type equations (NLSE) and the extended continuous Heisenberg ferromagnet equation (HFE) to investigate how nonlocality properties of one system are inherited by the other. First, we consider the space curves induced by the nonlinear Schrödinger-type equations and its equivalent spin systems. Such space curves are governed by the Serret–Frenet equation (SFE) for three basis vectors. We also show that the equation for the third of the basis vectors coincides with the well-known integrable HFE and its generalization. Two other equations for the remaining two vectors give new integrable spin systems. Finally, we investigated the relation between the differential geometry of surfaces and integrable spin systems for the three basis vectors.

A nonparametric probabilistic method to enhance PGD solutions with data-driven approach, application to the automated tape placement process

A nonparametric method assessing the error and variability margins in solutions depicted in a separated form using experimental results is illustrated in this work. The method assess the total variability of the solution including the modeling error and the truncation error when experimental results are available. The illustrated method is based on the use of the PGD separated form solutions, enriched by transforming a part of the PGD basis vectors into probabilistic one. The constructed probabilistic vectors are restricted to the physical solution’s Stiefel manifold. The result is a real-time parametric PGD solution enhanced with the solution variability and the confidence intervals.

On the coupling of relativistic particle to gravity and Wheeler–DeWitt quantization

A system consisting of a point particle coupled to gravity is investigated. The set of constraints is derived. It was found that a suitable superposition of those constraints is the generator of the infinitesimal transformations of the time coordinate [Formula: see text] and serves as the Hamiltonian which gives the correct equations of motion. Besides that, the system satisfies the mass shell constraint, [Formula: see text], which is the generator of the worldline reparametrizations, where the momenta [Formula: see text], [Formula: see text], generate infinitesimal changes of the particle’s position [Formula: see text] in spacetime. Consequently, the Hamiltonian contains [Formula: see text], which upon quantization becomes the operator [Formula: see text], occurring on the right-hand side of the Wheeler–DeWitt equation. Here, the role of time has the particle coordinate [Formula: see text], which is a distinct concept than the spacetime coordinate [Formula: see text]. It is also shown how the ordering ambiguities can be avoided if a quadratic form of the momenta is cast into the form that instead of the metric contains the basis vectors.

The Dual Graph Shift Operator: Identifying the Support of the Frequency Domain

Contemporary data is often supported by an irregular structure, which can be conveniently captured by a graph. Accounting for this graph support is crucial to analyze the data, leading to an area known as graph signal processing (GSP). The two most important tools in GSP are the graph shift operator (GSO), which is a sparse matrix accounting for the topology of the graph, and the graph Fourier transform (GFT), which maps graph signals into a frequency domain spanned by a number of graph-related Fourier-like basis vectors. This alternative representation of a graph signal is denominated the graph frequency signal. Several attempts have been undertaken in order to interpret the support of this graph frequency signal, but they all resulted in a one-dimensional interpretation. However, if the support of the original signal is captured by a graph, why would the graph frequency signal have a simple one-dimensional support? Departing from existing work, we propose an irregular support for the graph frequency signal, which we coin dual graph. A dual GSO leads to a better interpretation of the graph frequency signal and its domain, helps to understand how the different graph frequencies are related and clustered, enables the development of better graph filters and filter banks, and facilitates the generalization of classical SP results to the graph domain.