N-dimensional Problem Research Articles

Accelerating Globally Optimal Consensus Maximization in Geometric Vision.

Branch-and-bound-based consensus maximization stands out due to its important ability of retrieving the globally optimal solution to outlier-affected geometric problems. However, while the discovery of such solutions caries high scientific value, its application in practical scenarios is often prohibited by its computational complexity growing exponentially as a function of the dimensionality of the problem at hand. In this work, we convey a novel, general technique that allows us to branch over an n-1 dimensional space for an n-dimensional problem. The remaining degree of freedom can be solved globally optimally within each bound calculation by applying the efficient interval stabbing technique. While each individual bound derivation is harder to compute owing to the additional need for solving a sorting problem, the reduced number of intervals and tighter bounds in practice lead to a significant reduction in the overall number of required iterations. Besides an abstract introduction of the approach, we present applications to four fundamental geometric computer vision problems: camera resectioning, relative camera pose estimation, point set registration, and rotation and focal length estimation. Through our exhaustive tests, we demonstrate significant speed-up factors at times exceeding two orders of magnitude, thereby increasing the viability of globally optimal consensus maximizers in online application scenarios.

A Novel Evolutionary Algorithm: One-Dimensional Subspaces Optimization Algorithm (1D-SOA)

This paper introduces an evolutionary algorithm for n-dimensional single objective optimization problems: One-Dimensional Subspaces Optimization Algorithm (1D-SOA). The algorithm starts with an initial population in randomly selected positions. For each individual, a percentage of the total number of dimensions is selected, each dimension corresponding to a one-dimensional subspace. Later, it performs a symmetric search for the nearest local optima in all the selected one-dimensional subspaces (1D-S), for each individual at a time. The search stops if the new position does not improve the value of the objective function over all the selected 1D-S. The performance of the algorithm was compared against 11 algorithms and tested with 30 benchmark functions in 2 dimensions (D) and 30D. The proposed algorithm showed a better performance than all other studied algorithms for large dimensions.

Analysis of a radial free boundary tumor model with time-dependent absorption efficiency

We investigate an n-dimensional free boundary problem modeling tumor growth in radial form with angiogenesis process. This process is reflected in the Robin boundary condition where time-dependent absorption rate β(t) is considered. The system consists of a reaction-diffusion equation for the nutrient u(r,t) with nonlinear consumption and an integro-differential equation for the tumor radius R(t) with nonlinear cell proliferation. It is shown that the large-time behavior of the tumor is greatly tied to the properties of β(t). We prove that for any sufficiently small c>0, the tumor radius R(t) will remain bounded or shrink to zero when β(t) is uniformly bounded or tends to zero, respectively; and if β(t)→β⁎, the solution (u,R) will converge to the unique steady state (u⁎,R⁎) associating with β⁎. We also give examples of R(t) blowing up as t→∞ even if β(t)→0 when c is not small.

A reduction from an LWE problem to maximum independent set problems

The learning with errors (LWE) problem is a problem derived from machine learning that is believed to be intractable for quantum computers. This paper proposes a method that can reduce an LWE problem to a set of maximum independent set (MIS) problems, which are graph problems that are suitable for a quantum annealing machine to solve. The reduction algorithm can reduce an n-dimensional LWE problem to several small MIS problems with at most 2^{O(sqrt{n})} nodes when the lattice-reduction algorithm used in the LWE-reduction method successfully finds short vectors. The algorithm is useful for solving LWE problems in a quantum-classical hybrid manner by using an existing quantum algorithm to solve the MIS problems. For example, the smallest LWE challenge problem is reduced to MIS problems with about 40,000 vertices. This result means that the smallest LWE challenge problem will be within the scope of a real quantum computer in the near future.

Batch Gradient Learning Algorithm with Smoothing L1 Regularization for Feedforward Neural Networks

Regularization techniques are critical in the development of machine learning models. Complex models, such as neural networks, are particularly prone to overfitting and to performing poorly on the training data. L1 regularization is the most extreme way to enforce sparsity, but, regrettably, it does not result in an NP-hard problem due to the non-differentiability of the 1-norm. However, the L1 regularization term achieved convergence speed and efficiency optimization solution through a proximal method. In this paper, we propose a batch gradient learning algorithm with smoothing L1 regularization (BGSL1) for learning and pruning a feedforward neural network with hidden nodes. To achieve our study purpose, we propose a smoothing (differentiable) function in order to address the non-differentiability of L1 regularization at the origin, make the convergence speed faster, improve the network structure ability, and build stronger mapping. Under this condition, the strong and weak convergence theorems are provided. We used N-dimensional parity problems and function approximation problems in our experiments. Preliminary findings indicate that the BGSL1 has convergence faster and good generalization abilities when compared with BGL1/2, BGL1, BGL2, and BGSL1/2. As a result, we demonstrate that the error function decreases monotonically and that the norm of the gradient of the error function approaches zero, thereby validating the theoretical finding and the supremacy of the suggested technique.

Global classical solutions to a higher-dimensional doubly haptotactic cross-diffusion system modeling oncolytic virotherapy

This paper deals with the oncolytic virotherapy model{ut=DuΔu−ξu∇⋅(u∇v)+μuu(1−u)−ρuuz,x∈Ω,t>0,vt=−(αuu+αww)v+μvv(1−v),x∈Ω,t>0,wt=DwΔw−ξw∇⋅(w∇v)−δww+ρwuz,x∈Ω,t>0,zt=DzΔz−δzz−ρzuz+βw,x∈Ω,t>0, which was initially proposed by Alzahrani-Eftimie-Trucu (2019) [1] to model the process of oncolytic viral therapy, which is a therapeutic approach for cancer treatment. Here, Ω⊆RN(N≥1) is a bounded smooth domain with zero-flux boundary conditions, where Du,Dw,Dz,δw,δz,μu,ρu are positive parameters and ξu,αu,ξw,ρw,ρz,μv are nonnegative constants. The main results assert the global boundedness of solutions to an associated spatially N-dimensional initial-boundary value problems under suitably assumptions on the system parameters. We develop (refine) some integral estimate techniques and thereby improve previous results in [18,23,14]. To the best of our knowledge, there are the first results on boundedness of the system in higher dimension (N≥2).

Discrete interferences optimum beamformer in correlated signal and interfering noise

This paper introduces a significant special situation where the noise is a collection of D-plane interference signals and the correlated noise of D+1 is less than the number of array components. An optimal beamforming processor based on the minimum variance distortionless response (MVDR) generates and combines appropriate statistics for the D+1 model. Instead of the original space of the N-dimensional problem, the interference signal subspace is reduced to D+1. Typical antenna arrays in many modern communication networks absorb waves generated from multiple point sources. An analytical formula was derived to improve the signal to interference and noise ratio (SINR) obtained from the steering errors of the two beamformers. The proposed MVDR processor-based beamforming does not enforce general constraints. Therefore, it can also be used in systems where the steering vector is compromised by gain. Simulation results show that the output of the proposed beamformer based on the MVDR processor is usually close to the ideal state within a wide range of signal-to-noise ratio and signal-to-interference ratio. The MVDR processor-based beamformer has been experimentally evaluated. The proposed processor-based MVDR system significantly improves performance for large interference white noise ratio (INR) in the sidelobe region and provide an appropriate beam pattern.

Formula omitted]-Kernel-free soft quadratic surface support vector regression

In this paper, we propose a new regression method called the ∊-kernel-free soft quadratic surface support vector regression (∊-SQSSVR). After converting the n-dimensional regression problem into the (n+1)-dimensional classification problem, the principle of maximizing the sum of relative geometrical margin of each training point is used to construct our optimization problem, where the quadratic surface is restricted to be a hyperparaboloid by setting both the (n+1)-th row and (n+1)-th column of the corresponding matrix to be zero. The existence and uniqueness of the optimal solution to both primal and dual problems are also addressed. It should be pointed out that our model is nonlinear and kernel-free, so it does not need to select kernel function and corresponding parameters. At the same time, it is highly interpretable. In addition, our model is still a quadratic convex programming problem similar to the standard SVR. To visualize the effectiveness of our ∊-SQSSVR, 6 artificial datasets and 15 benchmark datasets are implemented in numerical experiments. The results show that our method is less time-consuming and as good as the nonlinear standard SVR with kernel function in comprehensive performances.

Criterion for the functional dissipativity of the Lamé operator

After introducing the concept of functional dissipativity of the Dirichlet problem in a domain Ω⊂RN for systems of partial differential operators of the form ∂h(Ahk(x)∂k) (Ahk(x) being m×m matrices with complex valued L∞ entries), we find necessary and sufficient conditions for the functional dissipativity of the two-dimensional Lamé system. As an application of our theory we provide two regularity results for the displacement vector in the N-dimensional equilibrium problem, when the body is fixed along its boundary.

Explainable fuzzy cluster-based regression algorithm with gradient descent learning

We propose an algorithm for n-dimensional regression problems with continuous variables. Its main property is explainability, which we identify as the ability to understand the algorithm’s decisions from a human perspective. This has been achieved thanks to the simplicity of the architecture, the lack of hidden layers (as opposed to deep neural networks used for this same task) and the linguistic nature of its fuzzy inference system. First, the algorithm divides the joint input-output space into clusters that are subsequently approximated using linear functions. Then, we fit a Cauchy membership function to each cluster, therefore identifying them as fuzzy sets. The prediction of each linear regression is merged using a Takagi-Sugeno-Kang approach to generate the prediction of the model. Finally, the parameters of the algorithm (those from the linear functions and Cauchy membership functions) are fine-tuned using Gradient Descent optimization. In order to validate this algorithm, we considered three different scenarios: The first two are simple one-input and two-input problems with artificial data, which allow visual inspection of the results. In the third scenario we use real data for the prediction of the power generated by a Combined Cycle Power Plant. The results obtained in this last problem (3.513 RMSE and 2.649 MAE) outperform the state of the art (3.787 RMSE and 2.818 MAE).

Rapid and safe wire tension distribution scheme for redundant cable-driven parallel manipulators

AbstractA non-iterative analytical approach is investigated to plan the safe wire tension distribution along with the cables in the redundant cable-driven parallel robots. The proposed algorithm considers not only tracking the desired trajectory but also protecting the system against possible failures. This method is used to optimize the non-negative wire tensions through the cables which are constrained based on the workspace conditions. It also maintains both actuators’ torque and cables’ tensile strength boundary limits. The pseudo-inverse problem solution leads to an n-dimensional convex problem, which is related to the robot degrees of redundancy. In this paper, a comprehensive solution is presented for a 1–3 degree(s) of redundancy in wire-actuated robots. To evaluate the effectiveness of this method, it is verified through an experimental study on the RoboCab cable robot in the infinity trajectory tracking task. As a matter of comparison, some standard methods like Active-set and sequential quadratic programming are also presented and the average elapsed time for each method is compared to the proposed algorithm.

An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional Space Fractional Wave and Damped Equations

In this paper, an efficient localized meshless method based on the space–time Gaussian radial basis functions is discussed. We aim to deal with the left Riemann–Liouville space fractional derivative wave and damped wave equation in high-dimensional space. These significant problems as anomalous models could arise in several research fields of science, engineering, and technology. Since an explicit solution to such equations often does not exist, the numerical approach to solve this problem is fascinating. We propose a novel scheme using the space–time radial basis function with advantages in time discretization. Moreover this approach produces the (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Therefore the local feature, as a remarkable and efficient property, leads to a sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. Some benchmark problems for wave models, both wave and damped, have been considered, highlighting the proposed method performances in terms of accuracy, efficiency, and speed-up. The obtained experimental results show the computational capabilities and advantages of the presented algorithm.

Global bifurcation for N-dimensional p-Laplacian problem and its applications

In this paper, we are concerned with the global bifurcation results for quasilinear elliptic problem where λ is a parameter, . Let B be a unit open ball of with a smooth boundary . We shall show that there are two distinct unbounded continua and , consisting of the bifurcation branch if f is not necessarily differentiable at the origin with respect to , and there are two distinct unbounded continua and , consisting of the bifurcation branch if f is not necessarily differentiable at infinity with respect to . As the applications of the above result, we shall prove more details about the existence and multiplicity results of sign-changing solutions for the elliptic problem where and g is not necessarily differentiable at the origin and infinity with respect to . Furthermore, by using a comparison theorem, we also obtain a non-existence result of nodal solutions to the above problem.

Group-Theoretic Buckling Analysis of Symmetric Plane Frames

This paper presents a computationally efficient formulation based on group theory, for the buckling analysis of symmetric plane frames. Numerical examples ranging from simple frames with one vertical plane of symmetry to more complex configurations with multiple symmetry planes are considered. In contrast to conventional procedures for accounting for symmetry, group theory uses the full symmetry of the configuration, resulting in higher reductions of computational effort. Typically, the n-dimensional problem is decomposed into smaller problems that are independent of each other, permitting the buckling loads and mode shapes to be computed through the solution of problems of much smaller dimension. The approach also yields insights on the character of the buckling modes before detailed computations are carried out.

An O(N) algorithm for computing expectation of N-dimensional truncated multi-variate normal distribution I: fundamentals

In this paper, we present the fundamentals of a hierarchical algorithm for computing the N-dimensional integral \(\phi (\mathbf {a}, \mathbf {b}; A) = {\int \limits }_{\mathbf {a}}^{\mathbf {b}} H(\mathbf {x}) f(\mathbf {x} | A) \text {d} \mathbf {x}\) representing the expectation of a function H(X) where f(x|A) is the truncated multi-variate normal (TMVN) distribution with zero mean, x is the vector of integration variables for the N-dimensional random vector X, A is the inverse of the covariance matrix Σ, and a and b are constant vectors. The algorithm assumes that H(x) is “low-rank” and is designed for properly clustered X so that the matrix A has “low-rank” blocks and “low-dimensional” features. We demonstrate the divide-and-conquer idea when A is a symmetric positive definite tridiagonal matrix and present the necessary building blocks and rigorous potential theory–based algorithm analysis when A is given by the exponential covariance model. The algorithm overall complexity is O(N) for N-dimensional problems, with a prefactor determined by the rank of the off-diagonal matrix blocks and number of effective variables. Very high accuracy results for N as large as 2048 are obtained on a desktop computer with 16G memory using the fast Fourier transform (FFT) and non-uniform FFT to validate the analysis. The current paper focuses on the ideas using the simple yet representative examples where the off-diagonal matrix blocks are rank 1 and the number of effective variables is bounded by 2, to allow concise notations and easier explanation. In a subsequent paper, we discuss the generalization of current scheme using the sparse grid technique for higher rank problems and demonstrate how all the moments of kth order or less (a total of O(Nk) integrals) can be computed using O(Nk) operations for k ≥ 2 and \(O(N \log N)\) operations for k = 1.

A Fractional Analysis in Higher Dimensions for the Sturm-Liouville Problem

Abstract In this work, we consider the n-dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the eigenvalues are real and simple. Moreover, using techniques from fractional variational calculus, we prove in the main result that the eigenvalues are separated and form an infinite sequence, where the eigenvalues can be ordered according to increasing magnitude. Finally, a connection with Clifford analysis is established.

General interval-valued overlap functions and interval-valued overlap indices

Overlap functions are aggregation functions that express the overlapping degree between two values. They have been used both as a conjunction in several practical problems (e.g., image processing and decision making), and to generate overlap indices between two fuzzy sets, which can be used to construct fuzzy confidence values to be applied in fuzzy rule based classification systems. Some generalizations of overlap functions were recently proposed, such as n-dimensional and general overlap functions, which allowed their application in n-dimensional problems. More recently, the concept of interval-valued overlap functions was presented, mainly to deal with uncertainty in providing membership functions. In this paper, we introduce: (i) the concept of n-dimensional interval-valued overlap functions, studying their representability, (ii) the definition of general interval-valued overlap functions, providing their characterization and some construction methods. Moreover, we also define the concept of interval-valued overlap index, as well as some constructing methods. In addition, we show an illustrative example where those new concepts are applied.

Chebyshev center of the intersection of balls: complexity, relaxation and approximation

We study the n-dimensional problem of finding the smallest ball enclosing the intersection of p given balls, the so-called Chebyshev center problem ( $${\mathrm{CC}}_{\mathrm{B}}$$ ). It is a minimax optimization problem and the inner maximization is a uniform quadratic optimization problem ( $$\mathrm{UQ}$$ ). When $$p\le n$$ , ( $$\mathrm{UQ}$$ ) is known to enjoy a strong duality and consequently ( $${\mathrm{CC}}_{\mathrm{B}}$$ ) is solved via a standard convex quadratic programming ( $$\mathrm{SQP}$$ ). In this paper, we first prove that ( $${\mathrm{CC}}_{\mathrm{B}}$$ ) is NP-hard and the special case when $$n=2$$ is polynomially solvable. With the help of a newly introduced linear programming relaxation (LP), the ( $$\mathrm{SQP}$$ ) relaxation is reobtained more directly and the first approximation bound for the solution obtained by ( $$\mathrm{SQP}$$ ) is established for the hard case $$p>n$$ . Finally, also based on (LP), we show that ( $${\mathrm{CC}}_{\mathrm{B}}$$ ) is polynomially solvable when either n or $$p-n(>0)$$ is fixed.

Pseudo Random Number Generation: a Reinforcement Learning approach

Abstract Pseudo-Random Numbers Generators (PRNGs) are algorithms produced to generate long sequences of statistically uncorrelated numbers, i.e. Pseudo-Random Numbers (PRNs). These numbers are widely employed in mid-level cryptography and in software applications. Test suites are used to evaluate PRNGs quality by checking statistical properties of the generated sequences. Machine learning techniques are often used to break these generators, i.e. approximating a certain generator or a certain sequence using a neural network. But what about using machine learning to generate PRNs generators? This paper proposes a Reinforcement Learning (RL) approach to the task of generating PRNGs from scratch by learning a policy to solve an N-dimensional navigation problem. In this context, N is the length of the period of the sequence to generate and the policy is iteratively improved using the average score of an appropriate test suite run over that period. Aim of this work is to demonstrate the feasibility of the proposed approach, to compare it with classical methods, and to lay the foundation of a research path which combines RL and PRNGs.

Analytic Solutions of the Eigenvalues of Mathieu’s Equation

Mathieu’s eigenvalue problem −y′′(x) + 2e_0 cos(2x)y(x) = λy(x), 0 < x < ℓ is symmetric if cos(2x) = cos(2ℓ − 2x) for ℓ = k0π, k0 ∈ N, and skew-symmetric if cos(2x) = − cos(2ℓ − 2x) for ℓ = π/2. Two typical boundary conditions are considered. When the eigenfunctions are expanded by the orthonormal bases of sine functions or cosine functions, we can derive an n-dimensional matrix eigenvalue problem, endowing with a special structure of the symmetric coefficient matrix A := [a_ij], a_ij = 0 if i + j is an odd integer. Based on it, we can obtain the eigenvalues easily and analytically. When ℓ = k_0π, k_0 ∈ N, we have a_ij = 0 if |i − j| > 2k_0. Besides the diagonal band, A has two off-diagonal bands, and furthermore, a cross band appears when k_0 ≥ 2. The product formula, the recursion formulas of characteristic functions and a fictitious time integration method (FTIM) are developed to find the eigenvalues of Mathieu’s equation.