Multivariate Chebyshev Research Articles

Optimization-aided construction of multivariate Chebyshev polynomials

This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a versatile semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all mononials up to degree six on the euclidean ball, the simplex, and the cross-polytope. Furthermore, inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before, namely for the monomial x12x22x3 on the euclidean ball and for the monomial x12x2x3 on the simplex.

Distribution-free Bayesian regularized learning framework for semi-supervised learning

In machine learning it is often necessary to assume or know the distribution of the data, however it is difficult to do so in practical applications. Aiming to this problem, this work, we propose a novel distribution-free Bayesian regularized learning framework for semi-supervised learning, which is called Hessian regularized twin minimax probability extreme learning machine (HRTMPELM). In this framework, we attempt to construct two non-parallel hyperplanes by introducing the high separation probability assumption, such that each hyperplane separates samples from one class with maximum probability while moving away from samples from the other class. Subsidiently, the framework can be utilized to construct reasonable semi-supervised classifiers by using the information of the inherent geometric distribution of the samples through the Hessian regularization term. Additionally, the proposed framework controls the misclassification error of samples by minimizing the upper limit of the worst-case misclassification probability, and improves the generalization performance of the model by introducing the idea of regularization to avoid the occurrence of ill-posedness and overfitting problems. More importantly, the framework has no hyperparameters, making the learning process very simplified and efficient. Finally, a simple and reliable algorithm with globally optimal solutions via multivariate Chebyshev inequalities is designed for solving the proposed learning framework. Experiments on multiple datasets demonstrate the reliability and effectiveness of the proposed learning framework compared to other methods. Especially, we applied the framework to Ningxia wolfberry quality detection, which greatly enriches and facilitates the application of machine learning algorithms in the agricultural field.

Uniqueness of solutions in multivariate Chebyshev approximation problems

We study the solution set to multivariate Chebyshev approximation problem, focussing on the ill-posed case when the uniqueness of solutions can not be established via strict polynomial separation. We obtain an upper bound on the dimension of the solution set and show that nonuniqueness is generic for ill-posed problems on discrete domains. Moreover, given a prescribed set of points of minimal and maximal deviation we construct a function for which the dimension of the set of best approximating polynomials is maximal for any choice of domain. We also present several examples that illustrate the aforementioned phenomena, demonstrate practical application of our results and propose a number of open questions.

Orthogonal Approximation of Invariant Manifolds in the Circular Restricted Three-Body Problem

Methods to parameterize and approximate the hyperbolic invariant manifolds of particular solutions in the circular restricted three-body problem (CR3BP) are presented in this paper. Analytical representations obtained from these manifold approximations are instrumental in the synthesis of optimal trajectories for cislunar transport. A multivariate Chebyshev series is used to approximate the surfaces, thereby serving as tractable parametric representations of the complex properties of motion. It is demonstrated that the continuum of ballistic capture trajectories and their associated sensitivities on the manifold can be realized in functional form using simple algebraic operations. Two applications making use of the Chebyshev manifold approximations as a terminal constraint surface are presented. The first is a low-thrust trajectory optimization problem formulated such that the optimal free final state lying on the manifold is determined as an additional set of design parameters. The second is a guidance law designed to target the manifold in the vicinity of the nominal patch point. Each of these methods takes advantage of the Chebyshev approximations to provide additional flexibility for mission design in multibody dynamic environments. These applications offer tremendous optimism about the utility of function approximation methods in arriving at a formal representation for the invariant manifolds in the three-body problem for efficient generation of optimal trajectories.

Stability and Stabilization of Polynomial Fuzzy-Model-Based Switched Nonlinear Systems Under MDADT Switching Signal

This article proposes a novel Lyapunov stability analysis for a class of switched nonlinear systems under the mode-dependent average dwell time (MDADT) switching signals. For the first time, a polynomial fuzzy model is used to describe the nonlinearity of the switched systems instead of a Takagi–Sugeno fuzzy model, which can represent a wider range of nonlinear plants. As existing advanced results are relatively conservative and lead to that tighter bounds on the dwell time are not easy to be obtained, therefore, the stability analysis for switched systems is still quite challenging. In this article, to tackle this problem, by developing a polynomial multiple Lyapunov function (MLF) approach and exploring the feature of the membership functions, relaxed stability conditions are derived for the switched polynomial fuzzy-model-based control systems. In particular, the polynomial MLF is guaranteed radially unbounded by a two-step procedure, making traditional quadratic MLF a specific case, thereby improved stability conditions can be established in the form of sum-of-squares. Moreover, the new algorithm for multivariate Chebyshev membership functions based on linear programming is proposed to introduce the information of membership functions and further relax the stability conditions to obtain tighter bounds on MDADT. Finally, a simulation example demonstrates the effectiveness of the developed techniques.

A Multivariate Chebyshev Bound of the Selberg Form

A Multivariate Chebyshev Bound of the Selberg Form

Multivariate approximation by polynomial and generalized rational functions

In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables.

Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials

Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function.

Pseudo-Lucas Functions of Fractional Degree and Applications

In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show possible applications of these new functions. For the first kind, we compute the fractional Newton sum rules of any orthogonal polynomial set starting from the entries of the Jacobi matrix. For the second kind, the representation formulas for the fractional powers of a r×r matrix, already introduced by using the pseudo-Chebyshev functions, are extended to the Lucas case.

Data Reconstruction in Heterogeneous Environmental Wireless Sensor Networks Using Robust Tensor Principal Component Analysis

The major challenges during the data acquisition process in an environmental wireless sensor network (EWSN) architecture are the presence of outliers and missing data. Outliers and missing data are ubiquitous in EWSN due to sensor failures, external noise, power dwindling, communication failures etc. Robust tensor principal component analysis (RTPCA) decomposes a noisy data tensor into a low-rank tensor and a sparse tensor, which can be exploited for the data recovery in EWSNs, where the low-rank component represents the intrinsic data tensor and the sparse component represents the gross outlier tensor. In this paper, a novel probabilistic outlier modelling scheme using multivariate Chebyshev's inequality hypothesis is proposed, which maps the sample population and the associated magnitudes of outliers with the spatio-temporal correlations in the acquired data. The inherent spatio-temporal and multi-attribute correlations in the EWSN data are established using singular value methods. A tensor nuclear norm (TNN) which extracts more temporal and multi-attribute correlations in the sensory data through block circulant matricization is used as the minimization function in RTPCA. In the forest surveillance scenario, usage of RTPCA results in a reconstruction accuracy of approximately 90% for a dataset with a high missing ratio of 0.95 and the outliers having largest possible magnitudes considered. For oceanographic data, in which the variance is lower, the simulations show similar performance for different outlier contaminations, although RPTCA performs better than other matrix and tensor data completion methods in terms of reconstruction accuracy.

Twin minimax probability machine for pattern classification

We propose a new distribution-free Bayes optimal classifier, called the twin minimax probability machine (TWMPM), which combines the benefits of both minimax probability machine(MPM) and twin support vector machine (TWSVM). TWMPM tries to construct two nonparallel hyperplanes such that each hyperplane separates one class samples with maximal probability, and is distant from the other class samples simultaneously. Moreover, the proposed TWMPM can control the misclassification error of samples in a worst-case setting by minimizing the upper bound on misclassification probability. An efficient algorithm for TWMPM is first proposed, which transforms TWMPM into concave fractional programming by applying multivariate Chebyshev inequality. Then the proposed TWMPM is reformulated as a pair of convex quadric programming (QP) by proper mathematical transformations. This guarantees TWMPM to have global optimal solution and be solved simply and effectively. In addition, we develop also an iterative algorithm for the proposed TWMPM. By comparing the two proposed algorithms theoretically, it is easy to know that the convex quadric programming algorithm is with lower computation burden than iterative algorithm for the TWMPM. A linear TWMPM version is first built, and then we show how to exploit mercer kernel to obtain nonlinear TWMPM version. The computation complexity for QP algorithm of TWMPM is in the same order as the traditional twin support vector machine (TWSVM). Experiments are carried out on three databases: UCI benchmark database, a practical application database and an artificial database. With low computation complexity and fewer parameters, experiments show the feasibility and effectiveness of the proposed TWMPM and its QP algorithm.

Composition Identities of Chebyshev Polynomials, via 2 × 2 Matrix Powers

Starting from a representation formula for 2 × 2 non-singular complex matrices in terms of 2nd kind Chebyshev polynomials, a link is observed between the 1st kind Chebyshev polinomials and traces of matrix powers. Then, the standard composition of matrix powers is used in order to derive composition identities of 2nd and 1st kind Chebyshev polynomials. Before concluding the paper, the possibility to extend this procedure to the multivariate Chebyshev and Lucas polynomials is touched on.

Improvements of the Markov and Chebyshev inequalities using the partial expectation

The multivariate Markov and multiple Chebyshev inequalities are improved using the partial expectation. The conditions of the strict inequalities between the improved and unimproved bounds of probabilities are given. The known upper bounds of the tail probabilities by the partial moment are derived as special cases of the Markov inequality, whose improved bounds and extensions to multivariate cases using the semicovariances and partial product moment are presented.

A minimax probability extreme machine framework and its application in pattern recognition

In this work we propose a minimax probability extreme learning machine framework (MPME), which combines the benefits of minimax probability machine (MPM) with extreme learning machine (ELM). For binary classification problems, we illustrate that the proposed MPME can be interpreted geometrically by minimizing the maximum of Mahalanobis distances to the two classes. Then two variants of the MPME are presented based on the l2-norm loss and l1-norm loss functions (called LSEMPME and LADMPME) respectively. Without making specific assumption on the data distribution, the proposed methods can provide explicit upper-bounds for the generalization error, moreover the LSEMPME and LADMPME minimize empirical risk simultaneously. The decision hyperplanes of the proposed methods pass through the origin in ELM feature space with few decision variables. By using the multivariate Chebyshev–Cantelli inequality, all the proposed problems can be reformulated as second-order cone programming (SOCP) with global solutions. Furthermore, numerical experiments have been carried out on two databases that are drawn from UCI benchmark database and a practical application database. First, the proposed methods are evaluated for a practical application consisting on the analysis of licorice seeds using near-infrared spectral (NIR) data. Experiments in six different spectral regions illustrate that the proposed methods can improve generalization in most cases. Then the proposed methods are evaluated on benchmark datasets. In comparison with traditional methods including MPM, ELM and support vector machine (SVM), experiments show that the proposed methods achieve comparable results in generalization. With few decision variables, the proposed methods are easy to implement for nonlinear classification and to estimate a lower-bound on the prediction accuracy.

The multivariate Markov and multiple Chebyshev inequalities

A sharp probability inequality named the multivariate Markov inequality is derived for the intersection of the survival functions for non-negative random variables as an extension of the Markov inequality for a single variable. The corresponding result in Chebyshev’s inequality is also obtained as a special case of the multivariate Markov inequality, which is called the multiple Chebyshev inequality to distinguish from the multivariate Chebyshev inequality for a quadratic form of standardized uncorrelated variables. Further, the results are extended to the inequalities for the union of the survival functions and those with lower bounds.

An Adaptive Partition of Unity Method for Multivariate Chebyshev Polynomial Approximations

Spectral polynomial approximation of smooth functions allows real-time manipulation of and computation with them, as in the Chebfun system. Extension of the technique to two-dimensional and three-d...

Multivariate polynomial interpolation on Lissajous–Chebyshev nodes

In this article, we study multivariate polynomial interpolation and quadrature rules on non-tensor product node sets related to Lissajous curves and Chebyshev varieties. After classifying multivariate Lissajous curves and the interpolation nodes linked to these curves, we derive a discrete orthogonality structure on these node sets. Using this orthogonality structure, we obtain unique polynomial interpolation in appropriately defined spaces of multivariate Chebyshev polynomials. Our results generalize corresponding interpolation and quadrature results for the Chebyshev–Gauß–Lobatto points in dimension one and the Padua points in dimension two.

Multivariate Chebyshev Inequality With Estimated Mean and Variance

ABSTRACTA variant of the well-known Chebyshev inequality for scalar random variables can be formulated in the case where the mean and variance are estimated from samples. In this article, we present a generalization of this result to multiple dimensions where the only requirement is that the samples are independent and identically distributed. Furthermore, we show that as the number of samples tends to infinity our inequality converges to the theoretical multi-dimensional Chebyshev bound.

Numerical constructions involving Chebyshev polynomials

We propose a new algorithm for the character expansion of tensor products of finite-dimensional irreducible representations of simple Lie algebras. The algorithm produces valid results for the algebras B3, C3, and D3. We use the direct correspondence between Weyl anti-invariant functions and multivariate second-kind Chebyshev polynomials. We construct the triangular trigonometric polynomials for the algebra D3.

Multivariate Chebyshev inequality in generalized measure space based on Choquet integral

ABSTRACTThe multivariate Chebyshev inequality for a random vector on probability measure space has been studied by numerous authors. One thing that seems missing is the multivariate version of Chebyshev inequality in non additive cases. In this article, we show that this inequality still works in generalized probability theory based on Choquet integral.