Local approximation of operators
Local approximation of operators
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DOI：
10.1016/j.acha.2023.01.004
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发表时间：
2022-02
期刊：
影响因子：
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通讯作者：
H. Mhaskar
中科院分区：
文献类型：
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作者：
H. Mhaskar
许多应用，如系统识别，时间序列的分类，偏微分方程的正问题和反问题，以及不确定性量化导致度量空间X和Y之间的非线性算子的逼近问题。我们研究了使用有限信息量确定此类算子在紧子集K X X上的逼近度的问题。如果F：KX → KY，对于某些F∈ KX，逼近F（F）的一个已建立的策略是将F（分别为F（F））编码为有限个真实的数d（分别为m）。再加上适当的重建算法（解码器），这个问题减少到一个紧凑的高维欧氏空间R d的子集上的m个函数的近似，等价地，嵌入在R d+ 1的单位球S d。这个问题是具有挑战性的，因为d，m，以及S d上的近似的复杂性都很大，并且有必要估计跟踪所有相关近似的相互依赖性的精度。在本文中，我们建立建设性的方法来做到这一点，有效地，即与常数参与的估计S d的近似为O（d 1/6）。我们研究了不同的光滑类的运营商，并提出了一种方法近似的F（F）只使用信息在一个小的邻域F，从而有效地减少所涉及的参数的数量。为了进一步缓解大量参数的问题，我们提出了预制网络，从而导致有效参数的数量大大减少。在确定性和概率设置的问题进行了研究。
Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces X and Y. We study the problem of determining the degree of approximation of such operators on a compact subset K X⊂ X using a finite amount of information. If F: K X→ K Y, a well established strategy to approximate F (F) for some F∈ K X is to encode F (respectively, F (F)) in terms of a finite number d (respectively m) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of m functions on a compact subset of a high dimensional Euclidean space R d, equivalently, the unit sphere S d embedded in R d+ 1. The problem is challenging because d, m, as well as the complexity of the approximation on S d are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; ie, with the constants involved in the estimates on the approximation on S d being O (d 1/6). We study different smoothness classes for the operators, and also propose a method for approximation of F (F) using only information in a small neighborhood of F, resulting in an effective reduction in the number of parameters involved. To further mitigate the problem of large number of parameters, we propose prefabricated networks, resulting in a substantially smaller number of effective parameters. The problem is studied in both deterministic and probabilistic settings.