Multivector Representation Research Articles

Improved Diversity-Promoting Collaborative Metric Learning for Recommendation.

Collaborative Metric Learning (CML) has recently emerged as a popular method in recommendation systems (RS), closing the gap between metric learning and collaborative filtering. Following the convention of RS, existing practices exploit unique user representation in their model design. This paper focuses on a challenging scenario where a user has multiple categories of interests. Under this setting, the unique user representation might induce preference bias, especially when the item category distribution is imbalanced. To address this issue, we propose a novel method called Diversity-Promoting Collaborative Metric Learning (DPCML), with the hope of considering the commonly ignored minority interest of the user. The key idea behind DPCML is to introduce a set of multiple representations for each user in the system where users' preference toward an item is aggregated by taking the minimum item-user distance among their embedding set. Specifically, we instantiate two effective assignment strategies to explore a proper quantity of vectors for each user. Meanwhile, a Diversity Control Regularization Scheme (DCRS) is developed to accommodate the multi-vector representation strategy better. Theoretically, we show that DPCML could induce a smaller generalization error than traditional CML. Furthermore, we notice that CML-based approaches usually require negative sampling to reduce the heavy computational burden caused by the pairwise objective therein. In this paper, we reveal the fundamental limitation of the widely adopted hard-aware sampling from the One-Way Partial AUC (OPAUC) perspective and then develop an effective sampling alternative for the CML-based paradigm. Finally, comprehensive experiments over a range of benchmark datasets speak to the efficacy of DPCML.

Improved Multiple Vector Representations of Images and Robust Dictionary Learning

Each sparse representation classifier has different classification accuracy for different samples. It is difficult to achieve good performance with a single feature classification model. In order to balance the large-scale information and global features of images, a robust dictionary learning method based on image multi-vector representation is proposed in this paper. First, this proposed method generates a reasonable virtual image for the original image and obtains the multi-vector representation of all images. Second, the same dictionary learning algorithm is used for each vector representation to obtain multiple sets of image features. The proposed multi-vector representation can provide a good global understanding of the whole image contour and increase the content of dictionary learning. Last, the weighted fusion algorithm is used to classify the test samples. The introduction of influencing factors and the automatic adjustment of the weights of each classifier in the final decision results have a significant indigenous effect on better extracting image features. The study conducted experiments on the proposed algorithm on a number of widely used image databases. A large number of experimental results show that it effectively improves the accuracy of image classification. At the same time, to fully dig and exploit possible representation diversity might be a better way to lead to potential various appearances and high classification accuracy concerning the image.

Ukrainian Diaspora Bandura Players Originates from Galicia

The article deals with the sociocultural origin of many well-known bandura art representatives of the Ukrainian diaspora. The work distinguishes the personalities of the diaspora bandura players, the natives of Galicia and their achievements in the context of the study necessity actualization in performing tradition broadcast, the preservation of the performing school characteristics, etc. The purpose of the article is to study the contribution of Galician bandura players to the development of bandura art in foreign countries. The scientific novelty of the research is the systematic information introduction about the representation of Galician bandura players in the Ukrainian diaspora, their priorities in the tools, repertoires, performances, and so on. The contribution of K. Misewych, D. Honta, D. Scherbyna to popularization of bandura in Poland in the interwar period is being determined. The activities of the Yuri Singalevich Lviv Galician school are being pointed out, whose students (S. Hanushevskyi, S. Lastovich- Chulivskyi, V. Yurkevych, Z. Shtokalko, etc.) not only promoted the bandura in Galicia, but also became successful in various types of bandura activity in the Ukrainian diaspora, synthesized their skills with traditional regional bandura schools. Particularly, the natives of Galicia V. Lutsiv and B. Shark, who in the Ukrainian diaspora synthesized the regional bandura schools of performance, were noted. The article uses historical, axiological, musicological and cultural approaches and their corresponding methods, which determine the methodology of research. In particular, the historical-chronological method is used to examine the stages of bandura players artistic path; the axiological - to determine the value of their achievements, musicological and cultural methods will contribute to the multi-vector representation of Galician bandura players contribution to the world treasury of Ukrainian culture. Conclusions. It has been determined, that in the process of bandura art development in Galicia, the unification of Western and Eastern bandura schools took place, the basic directions of music instrument functioning were laid, which ideas were continued in post-war emigration.

A dynamic evacuation simulation framework based on geometric algebra

Integrating dynamic analysis models into geographic information system (GIS)-based evacuation simulations is important yet complex. Different models must be smoothly assembled according to the data processing flow to obtain a dynamic, data-forced evacuation simulation. However, because of the diversity of data types and dynamic data updating among different models, closely integrated evacuation simulations are complex and inefficient. In this study, geometric algebra (GA) is introduced to develop a dynamic evacuation simulation framework for a hazardous gas diffusion scheme. In the framework, geospatial data are first integrated into a unified virtual scene with different forms of multivector representation. The major simulation models of gas diffusion, risk assessment, and dynamic evacuation routing compose the major steps of the evacuation simulation. On the basis of the generalized multivector structure, dynamic exchange and updating geospatial data at different evacuation steps can be performed seamlessly with the multivector structure and GA operators. The framework is tested with a case study of a three-dimensional residential area, which shows that our framework can support the integration of dynamic evacuation processes and the model integration is direct and smooth. This framework may also provide a new solution for the integration and dynamic data updating in spatiotemporal GIS.

Geometric algebra method for multidimensionally-unified GIS computation

Seamless multidimensional handling and coordinate-free characteristics of geometric algebra (GA) provide means to construct multidimensionally-unified GIS computation models. Using the multivector representation for basic geometric objects within GA, we are able to construct adaptable unified geometric-topological structural models of a multidimensional geographical scene. Multidimensional operators found within the geometry, topology and GIS analysis are developed with basic GA operators. A unified computational framework is proposed, it unifies expressions and operation structures, as well as supports the analysis of multidimensional complex scenes. Finally, we illustrate modelling a three-dimensional residential district, which shows that GA-based multidimensionally-unified computation models can effectively represent and analyze complex and multidimensional geographical scenes. The development of the proposed GIS multidimensionally-unified representation, analysis, and modeling enhances current GIS algorithms and geographical models.

The nullcone in the multi-vector representation of the symplectic group and related combinatorics

We study the nullcone in the multi-vector representation of the symplectic group with respect to a joint action of the general linear group and the symplectic group. By extracting an algebra over a distributive lattice structure from the coordinate ring of the nullcone, we describe a toric degeneration and standard monomial theory of the nullcone in terms of double tableaux and integral points in a convex polyhedral cone.

Parameter Estimation from Uncertain Data in Geometric Algebra

We show how standard parameter estimation methods can be applied to Geometric Algebra in order to fit geometric entities and operators to uncertain data. These methods are applied to three selected problems. One of which is the perspective pose estimation problem. We show experiments with synthetic data and compare the results of our algorithm with standard approaches. In general, our aim is to find multivectors that satisfy a particular constraint, which depends on a set of uncertain measurements. The specific problem and the type of multivector, representing a geometric entity or a geometric operator, determine the constraint. We consider the case of point measurements in Euclidian 3D-space, where the respective uncertainties are given by covariance matrices. We want to find a best fitting circle or line together with their uncertainty. This problem can be expressed in a linear manner, when it is embedded in the corresponding conformal space. In this space, it is also possible to evaluate screw motions and their uncertainty, in very much the same way. The parameter estimation method we use is a least-squares adjustment method, which is based on the so-called Gauss-Helmert model, also known as mixed model with constraints. For this linear model, we benefit from the implicit linearization when expressing our constraints in conformal space. The multivector representation of the entities we are interested in also allows their uncertainty to be expressed by covariance matrices. As a by-product, this method provides such covariance matrices.

Geometric preprocessing, geometric feedforward neural networks and Clifford support vector machines for visual learning

This paper shows the design and use of feed-forward neural networks and the support vector machines (SVM) in the coordinate-free mathematical system of the Clifford geometric algebra. We compare the McCulloch–Pitts neuron and the geometric neuron. An interesting case of the geometric neuron is the conformal neuron which can be used for RBF networks and SVM. The paper presents the generalization of the real- and complex-valued multilayer perceptron (MLP) to the Clifford valued multilayer perceptron. The paper studies also the multivector support vector machines (MSVM) which are SVMs for processing multivectors. For that we design kernels involving Clifford products. The resultant kernel resembles a sort of polynomial kernel using a multivector representation. In the context of SVMs an important contribution of the paper is the generalization of the real- and complex-valued SVM classifiers over the hyper-complex numbers. This Clifford valued SVM accepts multiple multivector inputs and it is a multi-class classifier. For the preprocessing the authors introduce a promising geometric method utilizing Clifford moments. This method is applied together with geometric MLPs for tasks of 2D pattern recognition. The experimental part shows applications of SVM using the conformal neuron and Clifford kernels. We include challenging applications of the Clifford SVM classifier for nonlinear separable problems. The authors believe that the use of the MLPs and SVMs in the geometric algebra framework expands their sphere of applicability for multivector learning in graded spaces.

A pictorial representation for multispin evolutions

AbstractModern NMR techniques involve keeping track of the evolutions of two or more coupled spins. A new multivector representation adequate for visualizing such evolutions is presented. This representation can be put in one‐one correspondence with product operators, and has the advantage that it retains all possible orientations of each spin in the product operator. Thus, as the spins evolve, faithful pictures can be drawn both to indicate the directional change of the spins magnetizations and to deduce the detailed signals that arise from their evolution. The results are in full agreement with density matrix calculations. In particular, one can visualize multiple‐quantum coherence transfer. Evolutions due to chemical shift, J coupling, and the effect of radio frequency (rf) pulses are pictured. A simple heteronuclear chemical shift correlation pulse sequence is used to illustrate how the multivector representation can follow through a combination of spin evolutions and how the signals can be correctly deduced entirely from pictures without the need to refer to density matrix or operator algebra calculations.