Approximate Vector Research Articles

A Novel PoW Scheme Implemented by Probabilistic Signature for Blockchain

PoW (Proof of Work) plays a significant role in most blockchain systems to grant an accounting right over decentralized participants and ensure tamper resistance. Though hash functions are generally exploited for PoW due to their merits on summering, anti-collision, and irreversibility, they cannot certify that the bookkeeper is exactly the worker. Thereafter, such insistence may lead to abuse or even embezzlement of computing power for the benefit of malicious miners. To preserve the functionality of PoW but also bind the miners’ signing keys with their works, we build a post-quantum PoW scheme by changing the approximate closest vector norm for probabilistic NTRUSign. Different from the schemes based on hash functions, our scheme takes signing as the proof of work where signature verification is just the evidence of block reward. We also presented a method to adjust the difficulty of signing by modifying the probability of generating a correct signature. The performance of our scheme is also analyzed theoretically and experimentally, which implies its practicability and advantages.

Measuring the Connectedness of the Global Economy

We develop a technique to exploit forecast error variance decompositions to evaluate the macroeconomic connectedness embedded in any multi-country macroeconomic model with an approximate vector autoregressive (VAR) representation. We apply our technique to a large global VAR model covering 25 countries and derive vivid representations of macroeconomic connectedness. We find that the US exerts a dominant influence in the global economy and that Brazil, China, and the Eurozone are also globally significant. Recursive analysis over the period of the global financial crisis shows that shocks to global equity markets are transmitted rapidly and forcefully to real trade flows and real GDP.

Education Development Issue in Activity of Kyiv City Council in 1906–1910s: Based on Information from Periodicals

The article considers the development of education in Kyiv in the early twentieth century. Particular attention is paid to the efforts of the City Council to build a network of educational institutions and implement the task of meeting the educational needs of the population of the suburbs and outskirts. It is revealed that the activity of the Duma, the cadence of 1906–1910s in the field of education was mainly limited to financial and economic affairs. Discussion of land allocation for the construction of educational institutions became the subject of sharp political discussions of vowels and became an obstacle to the rapid settlement of urgent educational processes. Through the efforts of the Duma of 1906–1910, the network of educational institutions was expanded to meet the needs of the growing city. During the cadence, 7 new single-grade schools were opened, a four-grade school named after Tsarevich Oleksiy and a women’s two-grade school. The Duma helped ensure access to education in all corners of the city, thanks to which the sixth gymnasium was opened in Lukyanivka, where until 1908 there were no educational institutions at all. Kyiv City Duma has repeatedly turned its attention to the maintenance of schools and their financial support. All of this required the flexibility of public decisions to settle the cost of school equipment and rent payments. The most important achievement was the development of a project of the city school network with a financial plan for general education with a implementation period of 12 years. A significant achievement was the census of school-age children. Based on the obtained statistical data, the school commission was able to make calculations and set an approximate vector of educational needs for Kyiv. The most informative sources that allow to cover in detail the decision-making process and the political struggle over the development of education are Kyiv periodicals, in particular, «Kyivlianyn» and «Rada».

On generalized approximate convex functions and variational inequalities

In this paper, we consider a vector optimization problem involving locally Lipschitz generalized approximately convex functions and provide several concepts of approximate efficient solutions. We formulate approximate vector variational inequalities of Minty and Stampacchia type under the framework of Clarke subdifferentials and use these inequalities as a tool to characterize an approximate efficient solution of the vector optimization problem.

Approximate Dynamic Programming Vector Controllers for Operation of IPM Motors in Linear and Overmodulation Regions

To improve the efficiency of a permanent magnet (PM) motor, it is usually required to extend the motor operation from linear to over or even six-step modulation regions based upon the space vector pulsewidth modulation technique. Traditionally, a PM motor is controlled by using the standard vector control technique. However, a recent study shows the decoupling inaccuracy of designing PM motor control using the conventional standard vector control approach. This issue has caused a challenge to control a PM motor, particularly when the motor operates from linear to overmodulation regions. In this article, a novel approximate dynamic programming (ADP) vector controller is developed to overcome the challenge. The ADP controller is developed using the full dynamic equation of a PM motor and implemented using an artificial neural network (ANN). A feedforward control strategy is integrated with the -based ADP controller to enhance the performance of the controller in both linear and overmodulation regions. The ANN-based ADP control is compared with the conventional standard vector control and a model predictive control. Simulation and hardware experiments demonstrate that the proposed controller can track reference changes better with high efficiency and reliability for PM motor operation in linear and overmodulation regions.

Augmented Subspaces in the LSQR Krylov Method

The LSQR iterative method is a Krylov subspace method for solving least-squares problems. Early termination is rare, and it is common for LSQR to require many iterations before an approximation of the solution with desired accuracy has been determined. In this paper, we present a restarted LSQR method and we use a new technique for accelerating the convergence of restated by adding some approximate error vectors to the Krylov subspace. The effectiveness of the new method is illustrated by several examples.

Video fingerprinting based on quadruplet convolutional neural network

In order to achieve fast and accurate retrieval of video copies, this paper proposes a compact video fingerprinting based on quadruplet convolutional neural network. The algorithm consists of four branch networks with shared weights, each branch network contains feature extraction and quantization coding. The projection and excitation network is combined with 3D convolution for feature extraction, which mainly learns feature weights to improve useful features and suppresses valueless features. The deep features learned are mapped into approximate real vectors in a fully connected form and quantized to generate binary codes. The model employs an improved quadruplet loss to divide the feature distance between the copied videos and the non-copied, and a quantization error term is added to ensure that the fingerprint codes contains as much similar information as the original data. The experimental results performed on the public dataset show that the algorithm can effectively improve the robustness and distinctiveness, and the average detection accuracy under multiple compound attacks is better than the compared algorithms.

Rebalancing the assembly lines with total squared workload and total replacement distance objectives

Assembly line balancing is an important and well recognised operations research problem. The current line balance may not stay optimal, even feasible, due to the disruptions in one or more workstations. In this study, after the disruption, we aim to rebalance the assembly line by considering the trade-off between workload balancing (fairness measure) and total replacement distance for the tasks assigned to the different workstations (stability measure). We try to generate all non-dominated objective function vectors for the defined fairness and stability measures. Two algorithms are developed: exact algorithm (classical approach) and tabu search algorithm. The results of the experiments have shown that the classical approach returns exact non-dominated objective vectors with up to 40 tasks and 7 workstations in one hour, and the tabu search algorithm returns approximate non-dominated objective vectors that are very close to their exact counterparts and can solve large sized instances with up to 94 tasks and 7 workstations in less than 10 s.

A Jacobi–Davidson Method for Large Scale Canonical Correlation Analysis

In the large scale canonical correlation analysis arising from multi-view learning applications, one needs to compute canonical weight vectors corresponding to a few of largest canonical correlations. For such a task, we propose a Jacobi–Davidson type algorithm to calculate canonical weight vectors by transforming it into the so-called canonical correlation generalized eigenvalue problem. Convergence results are established and reveal the accuracy of the approximate canonical weight vectors. Numerical examples are presented to support the effectiveness of the proposed method.

Laplace approximations for fast Bayesian inference in generalized additive models based on P-splines

Generalized additive models (GAMs) are a well-established statistical tool for modeling complex nonlinear relationships between covariates and a response assumed to have a conditional distribution in the exponential family. To make inference in this model class, a fast and flexible approach is considered based on Bayesian P-splines and the Laplace approximation. The proposed Laplace-P-spline model contributes to the development of a new methodology to explore the posterior penalty space by considering a deterministic grid-based strategy or a Markov chain sampler, depending on the number of smooth additive terms in the predictor. The approach has the merit of relying on a simple Gaussian approximation to the conditional posterior of latent variables with closed form analytical expressions available for the gradient and Hessian of the approximate posterior penalty vector. This enables to construct accurate posterior pointwise and credible set estimators for (functions of) regression and spline parameters at a relatively low computational budget even for a large number of smooth additive components. The performance of the Laplace-P-spline model is confirmed through different simulation scenarios and the method is illustrated on two real datasets.

Cryptosystem based on lattice and elliptic curve

In this work, we propose a new way to use lattice theory to build a public key cryptosystem and digital signature scheme. This cryptosystem based on the approximate closest vector problem and the problem of the discrete logarithm on an elliptic curve defined on a finite local ring. At first, we choose a point on the elliptic curve and we will make the exchange of keys to the Diffie-Hellman. We transform the coordinates of this point into a matrix which gives us the private key which will serve us for encryption and decryption.

Approximate solutions in nonsmooth and nonconvex cone constrained vector optimization

In this paper, we employ some advanced tools of variational analysis to provide new necessary optimality conditions for approximate (weak) Pareto solutions of a nonconvex and nonsmooth cone constrained vector optimization problem. The obtained necessary conditions are exhibited in a fuzzy form and a Fritz-John type. Sufficient optimality conditions for approximate (weak) Pareto solutions of the multiobjective problem are established by using assumptions of (strictly) approximately generalized convexity. Moreover, we address an approximate dual vector problem for the cone constrained vector optimization problem and examine converse and strong dualities for approximate (weak) Pareto solutions.

Analog Error-Correcting Codes

Coding schemes are presented that provide the ability to locate computational errors above a prescribed threshold while using analog resistive devices for approximate real vector–matrix multiplication. In such devices, the matrix is programmed into the device by setting an array of resistors to have conductances proportional to the respective entries in the matrix. In the coding scheme that is considered in this work, redundancy columns are appended so that each row in the programmed matrix forms a codeword of a prescribed linear code ${\mathcal {C}}$ over the real field; the result of the multiplication of any input real row vector by the matrix is then also a codeword of ${\mathcal {C}}$ . While error values within $\pm \delta $ in the entries of the result are tolerable (for some prescribed $\delta > 0$ ), outlying errors, with values outside the range $\pm \Delta $ (for a prescribed $\Delta \ge \delta $ ) should be located and corrected. As a design and analysis tool for such a setting, a certain functional is defined for the code ${\mathcal {C}}$ , through which a characterization is obtained for the number of outlying errors that can be handled, as a function of the ratio $\Delta /\delta $ . Several code constructions are then presented, primarily for the case of single outlying error handling. For this case, the coding problem is shown to be related to certain extremal problems on convex polygons.

Not-so-adiabatic quantum computation for the shortest vector problem

This paper describes adiabatic-inspired algorithms to solve the approximate shortest vector problem. Numerical simulations varying algorithm sweep-time suggest the existence of a Goldilocks zone for maximizing likelihood of success.

Sharp error bounds for Ritz vectors and approximate singular vectors

We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue problems. Using information that is available or easy to estimate, our bounds improve the classical Davis-Kahan $\sin\theta$ theorem by a factor that can be arbitrarily large, and can give nontrivial information even when the $\sin\theta$ theorem suggests that a Ritz vector might have no accuracy at all. We also present extensions in three directions, deriving error bounds for invariant subspaces, singular vectors and subspaces computed by a (Petrov-Galerkin) projection SVD method, and eigenvectors of self-adjoint operators on a Hilbert space.

An adaptive hybrid expansion method (AHEM) for efficient structural topology optimization under harmonic excitation

One challenge of solving topology optimization problems under harmonic excitation is that usually a large number of displacement and adjoint displacement vectors need to be computed at each iteration step. This work thus proposes an adaptive hybrid expansion method (AHEM) for efficient frequency response analysis even when a large number of excitation frequencies are involved. Assuming Rayleigh damping, a hybrid expansion for the displacement vector is developed, where the contributions of the lower-order modes and higher-order modes are given by the modal superposition and Neumann expansion, respectively. In addition, a simple (yet accurate) expression is derived for the residual error of the approximate displacement vector provided by the truncated hybrid expansion. The key factors affecting the convergence rate of the truncated hybrid expansion series are uncovered. Based on the Strum sequence, the AHEM can adaptively determine the number of lower-order eigenfrequencies and eigenmodes that need to be computed, while the number of terms that need to be preserved in the truncated Neumann expansion can be determined according to the given error tolerance. The performance of the proposed AHEM and its effectiveness for solving topology optimization problems under harmonic excitation are demonstrated by examining several 2D and 3D numerical examples. The non-symmetry of the optimum topologies for frequency response problems is also presented and discussed.

Approximation order of two-direction multiscaling functions

We investigate Condition E and the basic regularity conditions for the two-direction multiscaling functions $\boldsymbol {\phi }$, and then investigate approximation order of $\boldsymbol {\phi }$. By investigating a structure of approximation vectors, we are able to find simple and efficient criteria for Condition E, the basic regularity conditions, and the approximation order of $\boldsymbol {\phi }$. Examples for illustrating the general theory are given.

Provably Secure Identity-Based Encryption and Signature over Cyclotomic Fields

Identity-based cryptography is a type of public key cryptography with simple key management procedures. To our knowledge, till now, the existing identity-based cryptography based on NTRU is all over power-of-2 cyclotomic rings. Whether there is provably secure identity-based cryptography over more general fields is still open. In this paper, with the help of the results of collision resistance preimage sampleable functions (CRPSF) over cyclotomic fields, we give concrete constructions of provably secure identity-based encryption schemes (IBE) and identity-based signature schemes (IBS) based on NTRU over any cyclotomic field. Our IBE schemes are provably secure under adaptive chosen-plaintext and adaptive chosen-identity attacks, meanwhile, our IBS schemes are existentially unforgeable against adaptively chosen message and adaptively chosen identity attacks for any probabilistic polynomial time (PPT) adversary in the random oracle model. The securities of both schemes are based on the worst-case approximate shortest independent vectors problem (SIVPγ) over corresponding ideal lattices. The secret key size of our IBE (IBS) scheme is short—only one (two) ring element(s). The ciphertext (signature) is also short—only two (three) ring elements. Meanwhile, as the case of NTRUEncrypt, our IBE scheme could encrypt n bits in each encryption process. These properties may make our schemes have more advantages for some IoT applications over postquantum world in theory.

Structural sensitivity reanalysis formulations based on the polynomial-type extrapolation methods

This paper presents new structural sensitivity reanalysis formulations based on the polynomial-type extrapolation methods. In these formulations, the displacement vector of the modified structure is expressed in the form of the vector sequences based on the fixed-point iteration method. By using these vector sequences, the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE) methods calculate the approximate displacement vector of the modified structure by solving reduced linear least-square problems. Based on the definitions of the MPE and RRE methods, two sensitivity reanalysis formulations are derived, in which the first- and second-order sensitivities of the modified structure are obtained by solving a set of the over-determined least-square problems with much smaller size than the complete set of equations of the exact sensitivity analyses. The performance of the proposed sensitivity reanalysis formulations is evaluated by using four structural sensitivity reanalysis problems under multiple modifications in their initial designs. The results obtained from the numerical test problems indicate that the proposed sensitivity reanalysis formulations approximate the first- and second-order sensitivities of the modified structure with a high level of accuracy and they are able to converge to the exact solutions.

Dynamical systems: Approximate Lagrangians and Noether symmetries

We determine the approximate Noether point symmetries of the variational principle characterizing second-order equations of motion of a particle in a (finite-dimensional) Riemannian manifold. In particular, the Lagrangian comprises of kinetic energy and a potential [Formula: see text], perturbed to [Formula: see text]. We establish a convenient system of approximate geometric conditions that suffices for the computation of approximate Noether symmetry vectors and moreover, simplifies the problem of the effect of higher orders of the perturbation. The general results are applied to several practical problems of interest and we find extra Noether symmetries at [Formula: see text].