Local and Nonlinear Kernel Approximation

局部和非线性核逼近

• 批准号: 1232409
• 负责人: Thomas Hangelbroek
• 金额: $ 6.49万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-01-31 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1232409&HistoricalAwards=false
• 关键词: Local; Nonlinear; Kernel; Approximation

HangelbroekDMS-1047694 The goal of this project is to develop new tools andalgorithms for effective kernel approximation on Euclideandomains and certain compact manifolds. This work includes twoimportant aspects. The first is to develop schemes to treathighly nonuniform arrangements of data (with approximation ratescontrolled by a parameter reflecting the local density of thedata). The second is to devise nonlinear schemes thatapproximate using linear combinations of very few kernels. Schemes developed abide by two features of mainstreamapproximation theory (features that have generally been elusivefor kernel-based approximation schemes): they provideapproximation that is precise, by providing convergence ratesdictated by the smoothness of the approximand, and they areuniversal, by treating approximands at all levels of smoothness. The use of kernels to treat scattered, high-dimensional datais, by now, an established methodology in approximation theory. Kernels are especially prized for their ability to approximate inthe absence of underlying geometrical structures, like meshes ortriangulations. At this point there exist several algorithmsemploying kernels to treat large datasets that have been sampledalmost uniformly. However, the approximation power of suchalgorithms -- judged in terms of the fidelity of the approximantto the approximand -- is rarely completely understood. Furthermore, the question of how to treat highly nonuniform data(data with large gaps, or with points that coalesce) usingkernels is only beginning to be addressed. An important goal ofthis project is to develop kernel-based approximation methodsthat approximate from highly unstructured datasets and thatapproximate high-dimensional datasets with little computationaloverhead. Another goal is to acquire a precise understanding ofthe approximation power of such methods. Of particular interestare problems where there is some underlying geometric oralgebraic structure to be exploited, as, for example, is the casein problems in geodesy, crystallography, and molecular biology.

HangelbroekDMS-1047694 该项目的目标是开发新的工具和算法，以在欧几里得域和某些紧流形上进行有效的核逼近。 这项工作包括两个重要方面。 第一个是开发处理高度不均匀的数据排列的方案（由反映数据局部密度的参数控制的近似率）。 第二个是设计非线性方案，使用很少的内核的线性组合来近似。开发的方案遵循主流逼近理论的两个特征（对于基于核的逼近方案通常难以捉摸的特征）：它们通过提供由逼近数的平滑度决定的收敛率来提供精确的逼近，并且通过处理所有平滑度级别的逼近数，它们是通用的。 到目前为止，使用核来处理分散的高维数据是逼近论中的一种既定方法。内核因其在缺乏基础几何结构（例如网格或三角剖分）的情况下进行近似的能力而受到特别重视。 目前，存在几种使用内核来处理几乎均匀采样的大型数据集的算法。 然而，这种算法的逼近能力（根据近似值与近似值的保真度来判断）很少被完全理解。此外，如何使用核处理高度不均匀的数据（具有较大间隙或合并点的数据）的问题才刚刚开始得到解决。 该项目的一个重要目标是开发基于内核的近似方法，该方法可以从高度非结构化的数据集进行近似，并且可以以很少的计算开销来近似高维数据集。 另一个目标是准确理解此类方法的近似能力。 特别令人感兴趣的是需要利用一些潜在的几何或代数结构的问题，例如大地测量学、晶体学和分子生物学中的酪蛋白问题。