When exploring the broad application prospects of large-scale Gaussian process regression (GPR), three core challenges significantly constrain its full effectiveness: firstly, the O(n3) time complexity of computing th...
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When exploring the broad application prospects of large-scale Gaussian process regression (GPR), three core challenges significantly constrain its full effectiveness: firstly, the O(n3) time complexity of computing the inverse covariance matrix of n training points becomes an insurmountable performance bottleneck when processing large-scale datasets;Secondly, although traditional local approximation methods are widely used, they are often limited by the inconsistency of prediction results;The third issue is that many aggregation strategies lack discrimination when evaluating the importance of experts (i.e. local models), resulting in a loss of overall prediction accuracy. In response to the above challenges, this article innovatively proposes a comprehensive method that integrates third-degree stochastic fully symmetric interpolatory rules (TDSFSI), local approximation, and Tsallis mutual information (TDSFSIRLA), aiming to fundamentally break through existing limitations. Specifically, TDSFSIRLA first introduces an efficient third-degree stochastic fully symmetric interpolatory rules, which achieves accurate approximation of Gaussian kernel functions by generating adaptive dimensional feature maps. This innovation not only significantly reduces the number of required orthogonal nodes and effectively lowers computational costs, but also maintains extremely high approximation accuracy, providing a solid theoretical foundation for processing large-scale datasets. Furthermore, in order to overcome the inconsistency of local approximation methods, this paper adopts the Generalized Robust Bayesian Committee Machine (GRBCM) as the aggregation framework for local experts. GRBCM ensures the harmonious unity of the prediction results of each local model through its inherent consistency and robustness, significantly improving the stability and reliability of the overall prediction. More importantly, in response to the issue of uneven distribution of expert weights, this articl
In this study, motivating our earlier work [O. Duman and M.A. Ozarslan, Szasz-Mirakjan type operators providing a better error estimation. Appl. Math. Lett. 20, 1184-1188 (2007)], we investigate the local approximatio...
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In this study, motivating our earlier work [O. Duman and M.A. Ozarslan, Szasz-Mirakjan type operators providing a better error estimation. Appl. Math. Lett. 20, 1184-1188 (2007)], we investigate the local approximation properties of Szasz-Mirakjan type operators. The second modulus of smoothness and Petree's K-functional are considered in proving our results.
local parabolic splines on the axis R with equidistant nodes realizing the simplest local approximation scheme are considered. But, instead of the values of functions at the nodes, its formula approximates theirmean v...
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local parabolic splines on the axis R with equidistant nodes realizing the simplest local approximation scheme are considered. But, instead of the values of functions at the nodes, its formula approximates theirmean values in symmetric neighborhoods of these nodes. For an arbitrary averaging step H more than twice as large as the grid step h of the spline, the approximation errors are calculated exactly in the uniform metric of these functions and their derivatives for the function class W-infinity(2). For small averaging steps H <= 2h, these quantities were calculated by E. V. Strelkova (Shevaldina) in 2007.
Abstract: local approximation order to smooth complex valued functions by a finite dimensional space $\mathcal {H}$, spanned by certain products of exponentials by polynomials, is investigated. The results obt...
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Abstract: local approximation order to smooth complex valued functions by a finite dimensional space $\mathcal {H}$, spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline. The analysis of a typical space $\mathcal {H}$ is based here on the identification of its dual with a certain space $\mathcal {P}$ of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from $\mathcal {P}$, with interpolation data characterized by the structure of $\mathcal {H}$, and to construct bases of $\mathcal {P}$ corresponding to the interpolation problem.
We consider the problem of approximating the PageRank of a target node using only local information provided by a link server. We prove that local approximation of PageRank is feasible if and only if the graph has low...
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ISBN:
(纸本)9781605581644
We consider the problem of approximating the PageRank of a target node using only local information provided by a link server. We prove that local approximation of PageRank is feasible if and only if the graph has low in-degree and admits fast PageRank convergence. While natural graphs, such as the web graph, are abundant with high in-degree nodes, making local PageRank approximation too costly, we show that reverse natural graphs tend to have low indegree while maintaining fast PageRank convergence. It follows that calculating Reverse PageRank locally is frequently more feasible than computing PageRank locally. Finally, we demonstrate the usefulness of Reverse PageRank in five different applications.
The numerical manifold method falls into the category of the partition of unity methods. In order to enhance accuracy, high order polynomials can be specified as the local approximations. This, however, would incur ra...
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The numerical manifold method falls into the category of the partition of unity methods. In order to enhance accuracy, high order polynomials can be specified as the local approximations. This, however, would incur rank deficiency of the stiffness matrix. In this study, a local displacement approximation is constructed over a physical patch generated from a four quadrilateral mathematical mesh. All the degrees of freedom are physically meaningful. The stresses are continuous at all nodes, suggesting that no stress polish is required. The proposed approximations have the same accuracy as the first-order polynomials, but no linear dependency inherent in the latter. (C) 2016 Elsevier Ltd. All rights reserved.
Optimization techniques are useful tools to the design of complex systems. Especially in case of multiple conflicting performance indexes, the knowledge of the tradeoffs by means of Pareto optimality can help the desi...
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Optimization techniques are useful tools to the design of complex systems. Especially in case of multiple conflicting performance indexes, the knowledge of the tradeoffs by means of Pareto optimality can help the designer to achieve the best solution. Due to the increasing power of the computing tools, more and more accurate and time consuming models are used. In this case, the Pareto set computation can be a hard task (the Pareto set can be nonconvex, nonlinearities and discontinuities can occur) and the efficiency and the accuracy become crucial features for an optimization algorithm. In this paper an optimization algorithm based on local approximation of the objective and constraints functions is presented and tested with some well known test functions. The optimal design of the suspension system of a ground vehicle is performed by the new algorithm in order to reach the best tradeoff by means of road holding, comfort, working space and cornering behavior. The numerical results show that the proposed algorithm has good accuracy and high efficiency if compared to some widely used methods. The results are explained providing some general observations on the efficiency of local approximation based algorithm an other well known algorithms.
The recent years have witnessed a surge of interests of learning high-dimensional correspondence, which is important for both machine learning and neural computation community. Manifold learning-based researches have ...
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The recent years have witnessed a surge of interests of learning high-dimensional correspondence, which is important for both machine learning and neural computation community. Manifold learning-based researches have been considered as one of the most promising directions. In this paper, by analyzing traditional methods, we summarized a new framework for high-dimensional correspondence learning. Within this framework, we also presented a new approach, local approximation Maximum Variance Unfolding. Compared with other machine learning-based methods, it could achieve higher accuracy. Besides, we also introduce how to use the proposed framework and methods in a concrete application, cross-system personalization (CSP). Promising experimental results on image alignment and CSP applications are proposed for demonstration.
The problem of constructing a computer phasemeter using local approximation and nonlinear phase filtering algorithms is considered. The results of mathematical modeling, which illustrate the operation of the computer ...
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The problem of constructing a computer phasemeter using local approximation and nonlinear phase filtering algorithms is considered. The results of mathematical modeling, which illustrate the operation of the computer phasemeter algorithms, are presented.
We present algorithms for singular spectrum analysis and local approximation methods used to extrapolate time series. We analyze the advantages and disadvantages of these methods and consider the peculiarities of appl...
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We present algorithms for singular spectrum analysis and local approximation methods used to extrapolate time series. We analyze the advantages and disadvantages of these methods and consider the peculiarities of applying them to various systems. Based on this analysis, we propose a generalization of the local approximation method that makes it suitable for forecasting very noisy time series. We present the results of numerical simulations illustrating the possibilities of the proposed method.
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