Norm Of Vector Research Articles

Estimators for Structural Damage Detection Using Principal Component Analysis

Structural damage detection is an important issue in conservation. In this research, principal component analysis (PCA) has been applied to the temporal variation of modal frequencies obtained from a dynamic test of a scaled steel structure subjected to different damages and different temperatures. PCA has been applied in order to reduce, as much as possible, the number of variables involved in the problem of structural damage detection. The aim of the PCA study is to determine the minimum number of principal components necessary to explain all the modal frequency variation. Three estimators have been studied: T2 (the square of the vector norm of the projection in the principal component plan), Q (the square of the norm of the residual vector), and the variance explained. In the study, the results related to the undamaged structure needed one principal component to explain the modal frequency variation. However, the high damage configurations need five principal components to explain the modal frequency. The T2 and Q estimators have been arranged in order of increasing damage for all the performed experimental tests. The results indicate that these estimators could be useful to detect damage and to distinguish among a range of intensities of structural damage.

Event-triggered adaptive fuzzy control for stochastic nonlinear time-delay systems

This paper develops an event-triggered adaptive fuzzy control (AFC) scheme for stochastic nonlinear time-delay systems. With the aid of event-triggered mechanism, the salient superiority is to reduce the data transmission and communication burden. By virtue of Lyapunov-Krasovskii function, the time-delay term is effectively addressed. The fuzzy logic systems are exploited to model system uncertainties, meanwhile the norm of fuzzy weight vector is taken as estimation, which simplifies controller complexity. The devised AFC method can not only surmount the adverse influence deriving from time-delay, but also save network and communication resource. Finally, simulation results substantiate the validity of the designed control scheme.

Robust Self-Testing of Four-Qubit Symmetric States.

Quantum verification has been highlighted as a significant challenge on the road to scalable technology, especially with the rapid development of quantum computing. To verify quantum states, self-testing is proposed as a device-independent concept, which is based only on the observed statistics. Previous studies focused on bipartite states and some multipartite states, including all symmetric states, but only in the case of three qubits. In this paper, we first give a criterion for the self-testing of a four-qubit symmetric state with a special structure and the robustness analysis based on vector norm inequalities. Then we generalize the idea to a family of parameterized four-qubit symmetric states through projections onto two subsystems.

Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission

Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dual-polarized transmission as well as by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, to account for this fact, well-known lattice algorithms and related concepts are generalized in this work. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction to quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima—the norms of the shortest independent vectors of a lattice—and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized approaches outperform their real-valued counterparts in complexity and/or quality aspects. Moreover, the application of the generalized algorithms to MIMO communications is studied, particularly in the field of lattice-reduction-aided and integer-forcing equalization.

An Experimental Design Approach for Regret Minimization in Logistic Bandits

In this work we consider the problem of regret minimization for logistic bandits. The main challenge of logistic bandits is reducing the dependence on a potentially large problem dependent constant that can at worst scale exponentially with the norm of the unknown parameter vector. Previous works have applied self-concordance of the logistic function to remove this worst-case dependence providing regret guarantees that move the reduce the dependence on this worst case parameter to lower order terms with only polylogarithmic dependence on the main term and as well as linear dependence on the dimension of the unknown parameter. This work improves upon the prior art by 1) removing all scaling of the worst case term on the main term and 2) reducing the dependence on the dependence to scale with the square root of dimension in the fixed arm setting by employing an experimental design procedure. Our regret bound in fact takes a tighter instance (i.e., gap) dependent regret bound for the first time in logistic bandits. We also propose a new warmup sampling algorithm that can dramatically reduce the lower order term in the regret in general and prove that it can exponentially reduce the lower order term's dependency on the worst case parameter in some instances. Finally, we discuss the impact of the bias of the MLE on the logistic bandit problem in d dimensions, providing an example where d^2 lower order regret (cf., it is d for linear bandits) may not be improved as long as the MLE is used and how bias-corrected estimators may be used to make it closer to d.

Two incompatible types of invariants in the octonion spaces

This paper aims to study some invariants and conservation laws relevant to electromagnetic and gravitational fields by means of the rotational transformations of octonion coordinate systems. The scholars utilize the octonions to analyze the electromagnetic and gravitational fields simultaneously, including the octonion field potential, field strength, field source, linear momentum, angular momentum, torque and force. When the octonion coordinate system transforms rotationally, the vector part of one octonion may alter, while the scalar part of the octonion will remain unchanged. This property allows one to deduce a few invariants, such as the scalar part of octonion radius vector, speed of light, and norm of octonion radius vector. These invariants are the basic postulates for the Galilean transformation and Lorentz transformation. Furthermore, from the rotation transform of octonion coordinate systems, it is capable of deducing several invariants, including the mass, energy and power relevant to gravitational fields, in one octonion space [Formula: see text]. And the term relevant to the electric charge will transform with the rotation of coordinate systems. In another octonion space [Formula: see text], it is capable of inferring a few invariants, including the electric charge related to electromagnetic fields. And the terms relevant to the mass and energy will vary with the rotation of octonion coordinate systems. So, the invariants are divided into two different groups. In particular, the mass conservation law and energy conservation law can be effective simultaneously. But the charge conservation law and mass conservation law are unable to be valid simultaneously, in the strict sense. It is beneficial to further understand the laws of conservation.

Determination of “Neutral”–“Pain”, “Neutral”–“Pleasure”, and “Pleasure”–“Pain” Affective State Distances by Using AI Image Analysis of Facial Expressions

(1) Background: In addition to verbalizations, facial expressions advertise one’s affective state. There is an ongoing debate concerning the communicative value of the facial expressions of pain and of pleasure, and to what extent humans can distinguish between these. We introduce a novel method of analysis by replacing human ratings with outputs from image analysis software. (2) Methods: We use image analysis software to extract feature vectors of the facial expressions neutral, pain, and pleasure displayed by 20 actresses. We dimension-reduced these feature vectors, used singular value decomposition to eliminate noise, and then used hierarchical agglomerative clustering to detect patterns. (3) Results: The vector norms for pain–pleasure were rarely less than the distances pain–neutral and pleasure–neutral. The pain–pleasure distances were Weibull-distributed and noise contributed 10% to the signal. The noise-free distances clustered in four clusters and two isolates. (4) Conclusions: AI methods of image recognition are superior to human abilities in distinguishing between facial expressions of pain and pleasure. Statistical methods and hierarchical clustering offer possible explanations as to why humans fail. The reliability of commercial software, which attempts to identify facial expressions of affective states, can be improved by using the results of our analyses.

Understanding the effects of negative (and positive) pointwise mutual information on word vectors

ABSTRACT Despite the recent popularity of contextual word embeddings, static word embeddings still dominate lexical semantic tasks, making their study of continued relevance. A widely adopted family of such static word embeddings is derived by explicitly factorising the Pointwise Mutual Information (PMI) weighting of the co-occurrence matrix. As unobserved co-occurrences lead PMI to negative infinity, a common workaround is to clip negative PMI at 0. However, it is unclear what information is lost by collapsing negative PMI values to 0. To answer this question, we isolate and study the effects of negative (and positive) PMI on the semantics and geometry of models adopting factorisation of different PMI matrices. Word and sentence-level evaluations show that only accounting for positive PMI in the factorisation strongly captures both semantics and syntax, whereas using only negative PMI captures little of semantics but a surprising amount of syntactic information. Results also reveal that incorporating negative PMI induces stronger rank invariance of vector norms and directions, as well as improved rare word representations.

Observer‐based controller for nonaffine time‐delayed systems subject to input nonlinearities, state constraints, and unknown control direction

SUMMARYIn this work, an adaptive observer‐based control scheme has been designed for uncertain nonaffine nonstrict feedback systems subject to state time delay, various types of input nonlinearities, time‐varying asymmetric state constraints, and unknown control direction. Compared to the existing controllers for systems with state constraints, the designed control scheme in this work can be applied to state‐constrained systems with nonaffine structure subject to state delays, unknown control direction, and different types of input nonlinearities, while full‐states measurement is not required. Moreover, by introducing a novel saturated Nussbaum function in the present work, not only has the problem of unknown control direction been handled, but also the probability of a control shock due to applying the conventional Nussbaum functions has been eliminated. The system unknown dynamics has been approximated by using neural networks (NNs), and by updating the maximum norm of NNs weight vectors, only one adaptive law is required for the controller implementation. By employing appropriate Lyapunov functions, it has been shown that all signals in the closed‐loop system are bounded, state constraints are satisfied, and the output tracking error can be made small by selecting appropriate design parameters. The effectiveness of the designed control scheme has been shown through two simulation studies. Moreover, the superiority of the proposed saturated Nussbaum function in comparison with conventional Nussbaum functions has been shown via simulation.

Tamp-X: Attacking explainable natural language classifiers through tampered activations

While the technique of Deep Neural Networks (DNNs) has been instrumental in achieving state-of-the-art results for various Natural Language Processing (NLP) tasks, recent works have shown that the decisions made by DNNs cannot always be trusted. Recently Explainable Artificial Intelligence (XAI) methods have been proposed as a method for increasing DNN’s reliability and trustworthiness. These XAI methods are however open to attack and can be manipulated in both white-box (gradient-based) and black-box (perturbation-based) scenarios. Exploring novel techniques to attack and robustify these XAI methods is crucial to fully understand these vulnerabilities. In this work, we propose Tamp-X—a novel attack which tampers the activations of robust NLP classifiers forcing the state-of-the-art white-box and black-box XAI methods to generate misrepresented explanations. To the best of our knowledge, in current NLP literature, we are the first to attack both the white-box and the black-box XAI methods simultaneously. We quantify the reliability of explanations based on three different metrics—the descriptive accuracy, the cosine similarity, and the Lp norms of the explanation vectors. Through extensive experimentation, we show that the explanations generated for the tampered classifiers are not reliable, and significantly disagree with those generated for the untampered classifiers despite that the output decisions of tampered and untampered classifiers are almost always the same. Additionally, we study the adversarial robustness of the tampered NLP classifiers, and find out that the tampered classifiers which are harder to explain for the XAI methods, are also harder to attack by the adversarial attackers.

Robust security transmission design for multi-user peer-to-peer wireless relay networks

In this article, we studied the robust security transmission design for multi-user peer-to-peer relay networks, where all users demand secure communication and the eavesdropper is passive. Although the previous researches have designed the physical-layer security schemes under perfect channel state information, this study focuses on investigating the robust transmission design in the presence of a passive eavesdropper. Our goal is to maximize the artificial noise power to confuse the passive eavesdropper and subject to the worst-case signal-to-interference-noise-ratio constraints for all users under a bounded spherical region for the norm of the channel state information error vector from the relays to the destinations and the individual power constraints of all relay nodes. Mathematically, the original robust problem is difficult to solve due to its non-linearity and non-convexity. We propose to adopt S-Procedure and rank relaxation techniques to convert it to a semidefinite programming convex problem. The numerical results show the advantage of the proposed robust method.

Communication-Efficient Distributed SGD Using Random Access for Over-the-Air Computation

In this paper, we study communication-efficient distributed stochastic gradient descent (SGD) with data sets of users distributed over a certain area and communicating through wireless channels. Since the time for one iteration in the proposed approach is independent of the number of users, it is well-suited to scalable distributed SGD. Furthermore, since the proposed approach is based on preamble-based random access, which is widely adopted for machine-type communication (MTC), it can be easily employed for training models with a large number of devices where MTC is used for their connectivity. For fading channel, we show that noncoherent combining can be used. As a result, no channel state information (CSI) estimation is required. For over-the-air computation, the proposed approach takes advantage of random access, where the access probability is to encode the norm of local gradient vector (without using any additional bits) and the preamble is to encode the quantized normalized gradient vector. From analysis and simulation results, we can confirm that the proposed approach is not only scalable, but also provides improved performance as the number of devices increases. From analysis and simulation results, we can confirm that the proposed approach is not only scalable, but also provides improved performance as the number of devices increases.

Some Properties of Two Dimensional Interval Numbers

In this paper, we will introduce the notion of convergence of two dimensional interval sequences and show that the set of all two dimensional interval numbers is a metric space. Also, some ordinary vector norms will be extended to the set of two dimensional interval vectors. Furthermore, we will give definitions of statistical convergence, statistically Cauchy and Cesaro summability for the two dimensional interval numbers and we will get the relationships between them.

Method of Auxiliary Sources With an Excitation Source: The Auxiliary-Current Vector Norm

Method of Auxiliary Sources With an Excitation Source: The Auxiliary-Current Vector Norm

Orthogonality of invariant vectors

Let G be a finite group with given subgroups H and K. Let π be an irreducible complex representation of G such that its space of H-invariant vectors as well as the space of K-invariant vectors are both one dimensional. Let vH (resp. vK) denote an H-invariant (resp. K-invariant) vector of unit norm in a given G-invariant inner product 〈,〉π on π. Our interest is in computing the square of the absolute value of 〈vH,vK〉π. This is the correlation constant c(π;H,K) defined by Gross [3]. In this paper, we study this question for G=GL2(Fq), where Fq is the finite field of q=pm elements of odd characteristic p, H is its split torus and K is a non-split torus. The first main theorem of this paper gives an explicit formula for |〈vH,vK〉π|2 modulo p. The key idea here is to analyse the mod p reduction of π. The second main theorem relates the behaviour of 〈vH,vK〉π under Shintani base change and gives a sufficient condition for 〈vH,vK〉π to vanish for an irreducible representation π=BC(τ) of PGL2(E), in terms of the epsilon factor of the base changing representation τ of PGL2(F), where E/F is a finite extension of finite fields.

Three-Dimensional Kinematics during Shoulder Scaption in Asymptomatic and Symptomatic Subjects by Inertial Sensors: A Cross-Sectional Study.

Shoulder kinematics is a measure of interest in the clinical setting for diagnosis, evaluating treatment, and quantifying possible changes. The aim was to compare shoulder scaption kinematics between symptomatic and asymptomatic subjects by inertial sensors. Methods: Scaption kinematics of 27 subjects with shoulder symptomatology and 16 asymptomatic subjects were evaluated using four inertial sensors placed on the humerus, scapula, forearm, and sternum. Mobility, velocity, and acceleration were obtained from each sensor and the vector norm was calculated from the three spatial axis (x,y,Z). Shoulder function was measured by Upper Limb Functional Index and Disabilities of the Arm, Shoulder, and Hand questionnaires. One way ANOVA was calculated to test differences between the two groups. Effect size was calculated by Cohen’s d with 95% coefficient Intervals. Pearson’s correlation analysis was performed between the vector norms humerus and scapula kinematics against DASH and ULFI results in symptomatic subjects. Results: The asymptomatic group showed higher kinematic values, especially in the humerus and forearm. Symptomatic subjects showed significantly lower values of mobility for scapular protraction-retraction (Cohen’s d 2.654 (1.819–3.489) and anteriorisation-posteriorisation (Cohen’s d 1.195 (0.527–1.863). Values were also lower in symptomatic subjects for velocity in all scapular planes of motion. Negative correlation showed that subjects with higher scores in ULFI or DASH had lower kinematics values. Conclusion: Asymptomatic subjects tend to present greater kinematics in terms of mobility, velocity, and linear acceleration of the upper limb, and lower humerus and scapula kinematics in symptomatic subjects is associated with lower levels of function.

Consensus in the Hegselmann–Krause Model

This paper is concerned with the probability of consensus in a multivariate, spatially explicit version of the Hegselmann-Krause model for the dynamics of opinions. Individuals are located on the vertices of a finite connected graph representing a social network, and are characterized by their opinion, with the set of opinions $\Delta$ being a general bounded convex subset of a finite dimensional normed vector space. Having a confidence threshold $\tau$, two individuals are said to be compatible if the distance (induced by the norm) between their opinions does not exceed the threshold $\tau$. Each vertex $x$ updates its opinion at rate the number of its compatible neighbors on the social network, which results in the opinion at $x$ to be replaced by a convex combination of the opinion at $x$ and the nearby opinions: $\alpha$ times the opinion at $x$ plus $(1 - \alpha)$ times the average opinion of its compatible neighbors. The main objective is to derive a lower bound for the probability of consensus when the opinions are initially independent and identically distributed with values in the opinion set $\Delta$.

Analyzing radar rainfall estimate errors with three vector norms: application to weather radar rainfall data in Muğla, Turkey

Analyzing radar rainfall estimate errors with three vector norms: application to weather radar rainfall data in Muğla, Turkey

On the complexity of Putinar–Vasilescu's Positivstellensatz

We provide a new degree bound on the weighted sum-of-squares (SOS) polynomials for Putinar–Vasilescu's Positivstellensatz. This leads to another Positivstellensatz saying that if f is a polynomial of degree at most 2df nonnegative on a semialgebraic set having nonempty interior defined by finitely many polynomial inequalities gj(x)≥0, j=1,…,m with g1:=L−‖x‖22 for some L>0, then there exist positive constants c¯ and c depending on f,gj such that for any ε>0, for all k≥c¯ε−c, f has the decomposition(1)(1+‖x‖22)k(f+ε)=σ0+∑j=1mσjgj, for some SOS polynomials σj being such that the degrees of σ0,σjgj are at most 2(df+k). Here ‖⋅‖2 denotes the ℓ2 vector norm. As a consequence, we obtain a converging hierarchy of semidefinite relaxations for lower bounds in polynomial optimization on basic compact semialgebraic sets. The complexity of this hierarchy is O(ε−c) for prescribed accuracy ε>0. In particular, if m=L=1 then c=65, yielding the complexity O(ε−65) for the minimization of a polynomial on the unit ball.

Factoring isometries of quadratic spaces into reflections

Let V be a vector space endowed with a non-degenerate quadratic form Q. If the base field F is different from F2, it is known that every isometry can be written as a product of reflections. In this article, we detail the structure of the poset of all minimal length reflection factorizations of an isometry. If F is an ordered field, we also study factorizations into positive reflections, i.e., reflections defined by vectors of positive norm. We characterize such factorizations, under the hypothesis that the squares of F are dense in the positive elements (this includes Archimedean and Euclidean fields). In particular, we show that an isometry is a product of positive reflections if and only if its spinor norm is positive. As a final application, we explicitly describe the poset of all factorizations of isometries of the hyperbolic space.