{"title":"用高斯核进行多变量最坏情况逼近的渐近分析","authors":"A.A. Khartov , I.A. Limar","doi":"10.1016/j.jco.2024.101838","DOIUrl":null,"url":null,"abstract":"<div><p>We consider a problem of approximation of <em>d</em>-variate functions defined on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> which belong to the Hilbert space with tensor product-type reproducing Gaussian kernel with constant shape parameter. Within worst case setting, we investigate the growth of the information complexity as <span><math><mi>d</mi><mo>→</mo><mo>∞</mo></math></span>. The asymptotics are obtained for the case of fixed error threshold and for the case when it goes to zero as <span><math><mi>d</mi><mo>→</mo><mo>∞</mo></math></span>.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Asymptotic analysis in multivariate worst case approximation with Gaussian kernels\",\"authors\":\"A.A. Khartov , I.A. Limar\",\"doi\":\"10.1016/j.jco.2024.101838\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider a problem of approximation of <em>d</em>-variate functions defined on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> which belong to the Hilbert space with tensor product-type reproducing Gaussian kernel with constant shape parameter. Within worst case setting, we investigate the growth of the information complexity as <span><math><mi>d</mi><mo>→</mo><mo>∞</mo></math></span>. The asymptotics are obtained for the case of fixed error threshold and for the case when it goes to zero as <span><math><mi>d</mi><mo>→</mo><mo>∞</mo></math></span>.</p></div>\",\"PeriodicalId\":50227,\"journal\":{\"name\":\"Journal of Complexity\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-02-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Complexity\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0885064X24000153\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Complexity","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0885064X24000153","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们考虑的是定义在 Rd 上的 d 变量函数的近似问题，这些函数属于具有张量乘型再现高斯核且形状参数不变的希尔伯特空间。在最坏情况下，我们研究了信息复杂度随 d→∞ 的增长。在误差阈值固定的情况下，以及当误差阈值随 d→∞ 变为零时，我们得到了渐近线。
Asymptotic analysis in multivariate worst case approximation with Gaussian kernels
We consider a problem of approximation of d-variate functions defined on which belong to the Hilbert space with tensor product-type reproducing Gaussian kernel with constant shape parameter. Within worst case setting, we investigate the growth of the information complexity as . The asymptotics are obtained for the case of fixed error threshold and for the case when it goes to zero as .
期刊介绍：
The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited.
Areas Include:
• Approximation theory
• Biomedical computing
• Compressed computing and sensing
• Computational finance
• Computational number theory
• Computational stochastics
• Control theory
• Cryptography
• Design of experiments
• Differential equations
• Discrete problems
• Distributed and parallel computation
• High and infinite-dimensional problems
• Information-based complexity
• Inverse and ill-posed problems
• Machine learning
• Markov chain Monte Carlo
• Monte Carlo and quasi-Monte Carlo
• Multivariate integration and approximation
• Noisy data
• Nonlinear and algebraic equations
• Numerical analysis
• Operator equations
• Optimization
• Quantum computing
• Scientific computation
• Tractability of multivariate problems
• Vision and image understanding.
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