Kernel approximation with scalable bases

从空间中分散的地点、表面或更复杂的结构中获取的数据的近似和表示问题出现在不同的科学领域。核方法因其处理此类非结构化数据的能力而受到重视。它们也是无网格方法的重要例子，这种计算工具在不需要网格或三角测量等附带结构的情况下成功运行。首席研究员计划研究可扩展的局部基（一种处理核的强大的新方法）的分析和计算特性，并使用这些基来解决基于核的近似中的一些基本挑战。研究者希望生成算法，将使用的科学家和工程师的工作与非结构化数据和大规模数据集。在整个项目中，研究者将指导研究生参与本研究的理论和应用方面。与其他计算工具（如小波、样条和分段多项式有限元）不同，核方法不明确地涉及缩放操作：当数据变得更密集时，核不需要扩展。这允许核函数从复杂的几何构型中产生近似，其中单个尺度不明显。然而，它经常导致涉及大型，条件差的线性系统的问题，其解决既缓慢又不稳定。已经证明，在许多情况下，潜在的功能空间具有稳定的，高度局部化的基，可以快速构建。有令人信服的证据表明，许多计算问题的近线性处理是可能的，并且可能存在快速算法来处理涉及核的许多基本操作。提出的研究旨在在新的设置中生成可扩展的基（即，对于新的内核和新的流形），以理解和克服由边界效应和高度不均匀的数据排列引起的基本问题，并最终实现并行化，快速算法的核近似与这些基。

Problems involving the approximation and representation of data taken from scattered sites in space, from surfaces, or from more complicated structures arise in diverse scientific fields. Kernel methods are valued for their ability to treat such unstructured data. They are also important examples of meshless methods, computational tools that function successfully without the need of accompanying structures like grids or triangulations. The principal investigator plans to study analytic and computational properties of scalable localized bases (a powerful new methodology for working with kernels) and to use these bases to address some fundamental challenges in kernel based approximation. The investigator expects to generate algorithms that will be of use to scientists and engineers who work with unstructured data and large-scale datasets. Throughout the project, the investigator will mentor graduate students by involving them in both the theoretical and the applied aspects of this research.Unlike other computational tools (like wavelets, splines and piecewise polynomial finite elements), kernel methods do not explicitly involve a scaling operation: as data becomes more dense, the kernel is not required to be dilated. This allows kernels to produce approximates from complex geometrical configurations where a single scale is not apparent. However, it often leads to problems involving large, poorly conditioned linear systems, the solution of which is both slow and unstable. It has been demonstrated that in many cases the underlying function spaces possess stable, highly localized bases that can be constructed rapidly. There is compelling evidence that near linear processing of many computational problems is possible, and fast algorithms may exist to treat many of the basic operations involving kernels. The proposed research seeks to generate scalable bases in new settings (i.e., for new kernels and on new manifolds), to understand and overcome fundamental problems stemming from boundary effects and highly non-uniform arrangements of data, and ultimately to implement parallelized, fast algorithms for kernel approximation with these bases.