Euclidean Vector Space Research Articles

Quaternion Nuclear Norm Minus Frobenius Norm Minimization for color image reconstruction

Color image restoration methods typically represent images as vectors in Euclidean space or combinations of three monochrome channels. However, they often overlook the correlation between these channels, leading to color distortion and artifacts in the reconstructed image. To address this, we present Quaternion Nuclear Norm Minus Frobenius Norm Minimization (QNMF), a novel approach for color image reconstruction. QNMF utilizes quaternion algebra to capture the relationships among RGB channels comprehensively. By employing a regularization technique that involves nuclear norm minus Frobenius norm, QNMF approximates the underlying low-rank structure of quaternion-encoded color images. Theoretical proofs are provided to ensure the method’s mathematical integrity. Demonstrating versatility and efficacy, the QNMF regularizer excels in various color low-level vision tasks, including denoising, deblurring, inpainting, and random impulse noise removal, achieving state-of-the-art results.

On a generalized notion of metrics

In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2$$\\end{document} points. Some elementary properties of pseudo n-metrics are provided and their construction via exterior products is investigated. We discuss some examples from the geometry of Euclidean vector spaces leading to pseudo n-metrics on the unit sphere, on the Stiefel manifold, and on the Grassmann manifold. Further, we construct a pseudo n-metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n-metric.

Towards explainable interaction prediction: Embedding biological hierarchies into hyperbolic interaction space.

Given the prolonged timelines and high costs associated with traditional approaches, accelerating drug development is crucial. Computational methods, particularly drug-target interaction prediction, have emerged as efficient tools, yet the explainability of machine learning models remains a challenge. Our work aims to provide more interpretable interaction prediction models using similarity-based prediction in a latent space aligned to biological hierarchies. We investigated integrating drug and protein hierarchies into a joint-embedding drug-target latent space via embedding regularization by conducting a comparative analysis between models employing traditional flat Euclidean vector spaces and those utilizing hyperbolic embeddings. Besides, we provided a latent space analysis as an example to show how we can gain visual insights into the trained model with the help of dimensionality reduction. Our results demonstrate that hierarchy regularization improves interpretability without compromising predictive performance. Furthermore, integrating hyperbolic embeddings, coupled with regularization, enhances the quality of the embedded hierarchy trees. Our approach enables a more informed and insightful application of interaction prediction models in drug discovery by constructing an interpretable hyperbolic latent space, simultaneously incorporating drug and target hierarchies and pairing them with available interaction information. Moreover, compatible with pairwise methods, the approach allows for additional transparency through existing explainable AI solutions.

Multi-armed linear bandits with latent biases

In a linear stochastic bandit model, each arm corresponds to a vector in Euclidean space, and the expected return observed at each time step is determined by an unknown linear function of the selected arm. This paper addresses the challenge of identifying the optimal arm in a linear stochastic bandit model, where latent biases corrupt each arm's expected reward. Unlike traditional linear bandit problems, where the observed return directly represents the reward, this paper considers a scenario where the unbiased reward at each time step remains unobservable. This model is particularly relevant in situations where the observed return is influenced by latent biases that need to be carefully excluded from the learning model. For example, in recommendation systems designed to prevent racially discriminatory suggestions, it is crucial to ensure that the users' race does not influence the system. However, the observed return, such as click-through rates, may have already been influenced by racial attributes. In the case where there are finitely many arms, we develop a strategy to achieve O(|D|log⁡n) regret, where |D| is the number of arms and n is the number of time steps. In the case where each arm is chosen from an infinite compact set, our strategy achieves O(n2/3(log⁡n)1/2) regret. Experiments verify the efficiency of our strategy.

Retractions on closed sets

On a manifold or a closed subset of a Euclidean vector space, a retraction enables to move in the direction of a tangent vector while staying on the set. Retractions are a versatile tool to perform computational tasks such as optimization, interpolation, and numerical integration. This paper studies two known definitions of retraction on a closed subset of a Euclidean vector space, one being weaker than the other. Specifically, it shows that, in the context of constrained optimization, the weaker definition should be preferred as it inherits the main property of the other while being less restrictive.

On the Geometry of Weak Fuzzy Complex Numbers and Applications to the Classification of Some A-Curves

The concept of A-curves is considered as a novel application of real field extensions in solving some algebraic vectorial equations defined by Euclidean norms. In this paper, we present a novel insight through the classification of A-curves by illustrating many new semi-module isomorphisms between the direct product of weak fuzzy complex numbers with itself and the direct product of classical Euclidean vector spaces multiplied by itself. These isomorphisms will give us a full classification of A-curves that are related to weak fuzzy complex ring. Also, we provide many examples to explain the contribution of our work.

Advancing forensic research: An examination of compositional data analysis with an application on petrol fraud detection

In recent years, numerous studies have examined the chemical compounds of petrol and petrol data for forensic research. Standard quantitative methods often assume that the variables or compounds do not have compositional constraints or are not part of a constrained whole, operating within an Euclidean vector space. However, chemical compounds are typically part of a whole, and the appropriate vector space for their analysis is the simplex. Biased and arbitrary results result when statistical analysis are applied on such data without proper pre-processing of such data. Compositional analysis of data has not yet been considered in forensic science. Therefore, we compare classical statistical analysis as applied in forensic research and the new proposed paradigm of compositional data analysis (CoDa). It is demonstrated how such analysis improves the analysis in petrol and forensic science. Our study shows how principal component analysis (PCA) and classification results are affected by the preprocessing steps performed on the raw data.Our results indicate that results from a log ratio analysis provides a better separation between subgroups of the data and leads to an easier interpretation of the results. In addition, with a compositional analysis a higher classification accuracy is obtained. Even a non-linear classification method - in our case a random forest - was shown to perform poorly when applied without using compositional methods. Moreover, normalization of samples due to laboratory/unit-of-measurement effects is no longer necessary, since the composition of an observation is in compositional thinking equivalent to a multiple of it, because the used (log) ratios on raw and log ratio transformed data are equal.Petrol data from different petrol stations in Brazil are used for the demonstration. This data is highly susceptible to counterfeit petrol. Forensic analysis of its chemical elements requires non-biased statistical analysis designed for compositional data to detect fraud.Based on these results, we recommend the use of compositional data methods for gasoline and petrol chemical element analysis and gasoline product characterization, authentication and fraud detection in forensic sciences.

Multivariate phase space reconstruction and Riemannian manifold for sleep stage classification

Background and ObjectiveSleep was highly imperative in human daily life. However, an increasing number of people were undergoing sleep deprivation and sleep disorders. Sleep stage classification became a highly essential process in sleep scoring. Nevertheless, the visual scoring of arousals during sleep routinely performed by sleep specialists was a challenging exercise. MethodsThis paper introduced a novel approach for sleep stage classification using covariance feature matrix architecture with multivariate phase space reconstruction (MPSR). The goal was to capture the geometric properties and the hidden dynamic characteristics of multiple physiological signals. The covariance matrices constructed through the MPSR approach were considered as symmetric and positive definite (SPD) matrices, forming a Riemannian manifold space. These SPD matrices in the Riemannian manifold were mapped to the matrices in the tangent space, allowing them to be vectorized and treated as feature vectors in Euclidean space. Finally, an ensemble learning classifier was applied to perform various sleep stage tasks. ResultsOur proposed method was evaluated on three benchmark datasets to assess its effectiveness and robustness. For the tasks of both Five class and Six class sleep stages, the proposed approach in ten-fold cross validation achieved high accuracy of 93.57% and 92.56% for the Sleep EDF dataset, 86.36% and 84.18% for the DREAMS Subjects dataset, as well as 88.93% and 88.42% for the Sleep EDF Expanded dataset (95 subjects), respectively. In leave-one-subject-out cross validation, our proposed approach for the tasks of both Five class and Six class sleep stages yielded an accuracy of 84.46% and 80.73% for the Sleep EDF dataset, 82.50% and 79.51% for the DREAMS Subjects dataset, as well as 93.25% and 92.06% for the Sleep EDF Expanded-20 dataset (20 subjects), respectively. ConclusionCompared to the traditional sample covariance matrix (SCM), the covariance feature matrix using the MPSR method successfully captured the distinction of spatial information among various sleep stages. Moreover, our proposed method obtained good performance without requiring computationally large artifact suppression or a long signal decomposition process.

Computable Access Control: Embedding Access Control Rules Into Euclidean Space

Access control is one of the most basic techniques to ensure the security of the information system. The traditional access controls of information systems are usually performed based on the traversals or queries of rules. However, with the increasing complexity of information systems, massive data, and open environments bring great workload and risk for the traditional methods. This study proposes a model of embedding-based computable access control (ECAC), by employing the idea of representation learning in artificial intelligence. According to ECAC, access control rules can be embedded into a Euclidean vector space, and the security of arbitrary behavior can be computed by numerical vector operations, without any traditional querying or traversing of rules, and thus the workload of access control is reduced. Furthermore, by the embedding-based computation, the security of unknown behaviors can be predicted. Potentially, due to the use of numerical vectors instead of traditional semantic symbols, the risk of privacy leakage via semantics can be reduced. Finally, as the first embedding-based access control model, the effectiveness of ECAC is evaluated and concluded by the experiment-based analyses and discussions.

Estimating the relative importance of nodes in complex networks based on network embedding and gravity model

Identifying nodes that hold relative importance in complex networks is a burgeoning research area. Previous studies have demonstrated the effectiveness of the gravity model in capturing interactions among nodes in high-dimensional network space. Traditional algorithms predominantly concentrate on structural characteristics and node random walks in high-dimensional network space, overlooking substantial computational overhead and intricacies within the high-dimensional space. Simultaneously, the rapid evolution of the Internet has given rise to a series of emerging technologies, such as embedding models that can map high-dimensional network nodes to low-dimensional vector space. To address this issue, we introduce a novel algorithm for quantifying node relative importance, denoted as the Network Embedding and Gravity Model (NEGM). First, the network embedding method transforms nodes into low-dimensional, real-valued, dense vectors in Euclidean space. Then, drawing inspiration from Newton's law of universal gravitation, it proposes a novel gravity model. Finally, utilizing the novel gravity model, it calculates the aggregate attractive force of all nodes within the target node set. Experimental results show that NEGM excels in measuring the relative importance of nodes in various types of networks, demonstrating its significant potential for mining deep-seated information within authentic networks like protein networks and criminal networks.

High-Dimensional Approximate Nearest Neighbor Search: with Reliable and Efficient Distance Comparison Operations

Approximate K nearest neighbor (AKNN) search in the high-dimensional Euclidean vector space is a fundamental and challenging problem. We observe that in high-dimensional space, the time consumption of nearly all AKNN algorithms is dominated by that of the distance comparison operations (DCOs). For each operation, it scans full dimensions of an object and thus, runs in linear time wrt the dimensionality. To speed it up, we propose a randomized algorithm named ADSampling which runs in logarithmic time wrt the dimensionality for the majority of DCOs and succeeds with high probability. In addition, based on ADSampling we develop one generic and two algorithm-specific techniques as plugins to enhance existing AKNN algorithms. Both theoretical and empirical studies confirm that: (1) our techniques introduce nearly no accuracy loss and (2) they consistently improve the efficiency.

A characterization of the star duality mapping

We establish a characterization of the star duality mapping for star-shaped sets in n-dimensional Euclidean vector space.

Assessing Human Mobility by Constructing a Skeletal Database and Augmenting it Using a Generative Adversarial Network (GAN) Simulator.

This paper presents a neural network simulator based on anonymized patient motions that measures, categorizes, and infers human gestures based on a library of anonymized patient motions. There is a need for a sufficient training set for deep learning applications (DL). Our proposal is to extend a database that includes a limited number of videos of human physiotherapy activities with synthetic data. As a result of our posture generator, we are able to generate skeletal vectors that depict human movement. A human skeletal model is generated by using OpenPose (OP) from multiple-person videos and photographs. In every video frame, OP represents each human skeletal position as a vector in Euclidean space. The GAN is used to generate new samples and control the parameters of the motion. The joints in our skeletal model have been restructured to emphasize their linkages using depth-first search (DFS), a method for searching tree structures. Additionally, this work explores solutions to common problems associated with the acquisition of human gesture data, such as synchronizing activities and linking them to time and space. A new simulator is proposed that generates a sequence of virtual coordinated human movements based upon a script.

Quantitative characterization of several entanglement detection criteria

Quantitative characterization of different entanglement detection criteria for bipartite systems is presented. We review the implication sequence of these criteria and then numerically estimate volume ratios between criteria nonviolating quantum states and all quantum states. The numerical approach is based on the hit-and-run algorithm, which is applied to the convex set of all quantum states embedded into a Euclidean vector space of the Hilbert-Schmidt inner product. We demonstrate that reduction, majorization, and the R\'enyi-entropy-based criteria are very ineffective compared to the positive partial transpose. In the case of the R\'enyi-entropy-based criterion, we show that the ratio of detectable entanglement increases with the order of the R\'enyi entropy.

Open set recognition of underwater acoustic targets based on GRU-CAE collaborative deep learning network

Aiming at the problem of underwater acoustic target open set recognition, a new recognition method based on deep learning and template matching is proposed in this paper. The proposed method includes three stages: feature extraction based on GRU-CAE collaborative deep learning network, template establishment and template matching. Firstly, the GRU-CAE collaborative deep learning network is proposed to extract the deep collaborative features. The GRU-CAE cooperative deep learning network combines the advantages of CAE network in extracting spatial information of spectrogram and GRU network in extracting temporal structure features of spectrogram. Then, the feature template of each class of underwater acoustic target is constructed by finding an optimal vector in Euclidean space. Finally, the distance between the deep collaborative features of the test set and templates are calculated to distinguish the category of the samples. The experimental data of 5 kinds of underwater acoustic targets are used to verify the recognition performance of the proposed method. The results show that the deep collaborative features have excellent class separability, and the recognition accuracy rate of open set recognition can reach 82.21%. Compared with the other 3 open set recognition methods, the proposed method has the highest recognition accuracy rate.

On the divergence theorem for submanifolds of Euclidean vector spaces within the theory of second-gradient continua

In the theory of second-gradient continua, the internal virtual work functional can be considered as a second-order distribution in which the virtual displacements take the role of test functions. In its easiest representation, the internal virtual work functional is represented as a volume integral over a subset of the three-dimensional Euclidean vector space and involves first and second derivatives of the virtual displacements. In this paper, we show by an iterative integration by parts procedure how an alternative representation of such a functional can be obtained when the integration domain is a subset that contains also edges and wedges. Since this procedure strongly relies on the divergence theorem for submanifolds of a Euclidean vector space, it is a main goal to derive this divergence theorem for submanifolds starting from Stokes’ theorem for manifolds. To that end, results from Riemannian geometry are gathered and applied to the submanifold case.

Research on phase-locked loop technique based on three-dimensional coordinate transformation

The traditional phase-locked loop (PLL) technique widely used in three-phase four-wire system can only obtain one phase angle, and cannot obtain other parameters of unbalanced voltages, such as the amplitude and phase angle of each phase voltage. In order to improve the performance of PLL in unbalanced three-phase four-wire system with disturbance and reduce the order of PLL to improve the stability of the system, a PLL technique based on three-dimensional (3D) coordinate transformation was proposed in this paper. Firstly, the phase detector error of three-phase unbalanced voltage was deduced based on the 3D coordinate transformation. The error can be regarded as the expression form in Hilbert space after linearizing of it. Based on the correspondence between the orthonormal basis in Hilbert space and the unit vectors in Euclidean space, the error can be mapped to Euclidean space, and the detection control law for each voltage parameter can be designed respectively based on the principle of perpendicular decoupling. Then, the voltage parameter detection control law in Euclidean space is mapped back to Hilbert space, so the PLL technique scheme can be obtained which is beneficial to the implementation in the actual system. The control law of each unbalance indices, such as the amplitude and initial phase of each phase voltage, can be decoupled into a first-order or second-order system, so the system structure is simple, and the control parameter design is convenient. The detection of unbalance indices of the three-phase unbalanced voltages with zero-sequence component can be realized with fast and stable characteristics. The simulation and experiments of the proposed PLL technique are carried out finally under several transient conditions of grid voltages, stable within a quarter cycle at the fastest and always stable in 50 ms, so as to validate the rapidness and stability of the proposed PLL technique.

ПРЕДСТАВЛЕНИЕ ПАРНОГО ПОТЕНЦИАЛА ВЗАИМОДЕЙСТВИЯ В ВИДЕ МНОГОМЕРНЫХ РАЦИОНАЛЬНЫХ РЯДОВ В ПЕРЕМЕННЫХ ЯКОБИ ДЛЯ ЗАДАЧ МНОГИХ ТЕЛ

Scalar power functions of the form x1 + + xN -v Î are in some cases found in physical problems and applications, especially in many-body problems with paired interactions. There are known decompositions for two vectors in three-dimensional space. In this paper, we consider analogous decompositions with any number of N arbitrary M-dimensional vectors in Euclidean space as a product of a multidimensional rational series with respect to spatial variables and hyperspheric functions on the unit sphere SM-1. Such an advantage of expansion arises in three-body problems when solving the Faddeev equation, where it is known that the main problem is the approximate choice of approximation of interaction potentials, in which the t-matrix scattering elements acquired a separable form.

Convex bodies generated by sublinear expectations of random vectors

We show that many well-known transforms in convex geometry (in particular, centroid body, convex floating body, and Ulam floating body) are special instances of a general construction, relying on applying sublinear expectations to random vectors in Euclidean space. We identify the dual representation of such convex bodies and describe a construction that serves as a building block for all so defined convex bodies. Sublinear expectations are studied in mathematical finance within the theory of risk measures. In this way, tools from mathematical finance yield a whole variety of new geometric constructions.

SPECIAL VINBERG CONES

The paper is devoted to the generalization of the Vinberg theory of homogeneous convex cones. Such a cone is described as the set of “positive definite matrices” in the Vinberg commutative algebra ℋn of Hermitian T-matrices. These algebras are a generalization of Euclidean Jordan algebras and consist of n × n matrices A = (aij), where aii ∈ ℝ, the entry aij for i < j belongs to some Euclidean vector space (Vij ; \U0001d524) and {a}_{ji}={a}_{ij}^{ast }=mathfrak{g}left({a}_{ij},cdot right)in {V}_{ij}^{ast } belongs to the dual space {V}_{ij}^{ast }. The multiplication of T-Hermitian matrices is defined by a system of “isometric” bilinear maps Vij × Vjk → Vij ; i < j < k, such that |aij ⋅ ajk| = |aij| ⋅ |aik|, alm ∈ Vlm. For n = 2, the Hermitian T-algebra ℋn= ℋ2 (V) is determined by a Euclidean vector space V and is isomorphic to a Euclidean Jordan algebra called the spin factor algebra and the associated homogeneous convex cone is the Lorentz cone of timelike future directed vectors in the Minkowski vector space ℝ1,1⊕ V . A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra ℋ3(V; S) associated to a Clifford Cl(V )-module S together with an “admissible” Euclidean metric \U0001d524S.We generalize the construction of rank 2 Vinberg algebras ℋ2(V ) and special Vinberg algebras ℋ3(V; S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S0 ⊕ S1 is a ℤ2-graded Clifford Cl(V )-module with an admissible pseudo-Euclidean metric. The associated cone \U0001d4b1 is a homogeneous, but not convex cone in ℋm; m = 2; 3. We calculate the characteristic function of Koszul-Vinberg for this cone and write down the associated cubic polynomial. We extend Baez’ quantum-mechanical interpretation of the Vinberg cone \U0001d4b12 ⊂ ℋ2(V ) to the special rank 3 case.