CDS&E: Collaborative Research: Hierarchical Kernel Matrices for Scientific and Data Applications

机器学习和科学计算中的核矩阵描述了可以表示各种类型信息的点集合之间的关系。各种学科中数据集的规模不断增加，计算机硬件的计算能力不断提高，这使得我们的算法和核矩阵软件具有可扩展性，并且执行它们所需的时间随着问题的规模线性或接近线性增长。否则，这种大规模的数据问题可能无法处理。这个项目通过利用在这些矩阵中经常发现的层次结构来解决与处理内核矩阵相关的扩展瓶颈。通过加速核矩阵的计算，本研究可以在不确定性量化、积分方程问题、粒子模拟和地质统计学等不同领域进行大规模数据分析和科学模拟。实现新开发方法的高性能软件将在开源环境中开发。该项目专门解决高维问题，在机器学习中使用专门的核函数，以及为核矩阵构建分层表示的高初始计算成本。开发的新方法将应用于科学应用和机器学习应用中的大规模案例：布朗动力学和高斯过程回归。在机器学习中，新方法将补充高斯过程的现有大规模方法。高性能软件将在构建层次矩阵时解决特定的缩放挑战。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Kernel matrices in machine learning and scientific computing describe the relationships between collections of points which may represent various types of information. The increasing size of data sets in various disciplines and the increasing computational capability of computer hardware make it essential that our algorithms and software for kernel matrices are scalable, and that the time it takes for their execution grows linearly or close to linearly, with the problem size. Otherwise, such large-scale data problems may not be tractable. This project addresses the scaling bottlenecks associated with handling the kernel matrix by exploiting a hierarchical structure that is often found in these matrices. By accelerating computations with kernel matrices, this research enables large-scale data analysis and scientific simulation in diverse areas such as uncertainty quantification, integral equation problems, particle simulations, and geostatistics. High-performance software implementing the newly developed methods will be developed in an open-source environment.This project specifically addresses high-dimensional problems, the use of specialized kernel functions in machine learning, and the high initial computational cost of constructing a hierarchical representation for a kernel matrix. New methods developed will be applied to large-scale cases in a scientific application and a machine learning application: Brownian dynamics and Gaussian process regression. In machine learning, the new methods will complement existing large-scale approaches for Gaussian processes. High-performance software will address specific scaling challenges in constructing hierarchical matrices.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.