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

CDS

• 批准号: 2003720
• 负责人: Yuanzhe Xi
• 金额: $ 30.58万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-10-01 至 2024-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2003720&HistoricalAwards=false
• 关键词: CDS& Collaborative; Research; Hierarchical; Kernel

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.

机器学习和科学计算中的核矩阵描述了可能代表各种类型信息的点集合之间的关系。 在各种学科的数据集的大小和计算机硬件的计算能力的增加，使我们的算法和软件的核心矩阵是可扩展的，它所需要的时间线性增长或接近线性增长，与问题的大小是至关重要的。否则，这种大规模的数据问题可能无法处理。 这个项目解决了与处理内核矩阵相关的缩放瓶颈，方法是利用这些矩阵中经常出现的层次结构。 通过使用核矩阵加速计算，这项研究可以在不同领域进行大规模数据分析和科学模拟，例如不确定性量化，积分方程问题，粒子模拟和地质统计学。 将在开放源代码环境中开发实现新开发方法的高性能软件。该项目专门解决高维问题，在机器学习中使用专门的内核函数，以及为内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，新方法将补充现有的高斯过程的大规模方法。 高性能软件将解决在构建层次矩阵时遇到的特定扩展挑战。该奖项反映了NSF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。

项目成果

AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows

AUTM Flow：用于单调归一化流的原子无限制时间机

• 发表时间: 2022
• 期刊: Uncertainty in artificial intelligence
• 作者: Cai, D.;Ji, Y.;He, H.;Ye, Q.
• 通讯作者: Ye, Q.

Fast Deterministic Approximation of Symmetric Indefinite Kernel Matrices with High Dimensional Datasets

高维数据集对称不定核矩阵的快速确定性逼近

• DOI: 10.1137/21m1424627
• 发表时间: 2022
• 期刊: SIAM Journal on Matrix Analysis and Applications
• 影响因子: 1.5
• 作者: Cai, Difeng;Nagy, James;Xi, Yuanzhe
• 通讯作者: Xi, Yuanzhe

parGeMSLR: A parallel multilevel Schur complement low-rank preconditioning and solution package for general sparse matrices

parGeMSLR：用于一般稀疏矩阵的并行多级 Schur 补充低秩预处理和解决方案包

• DOI: 10.1016/j.parco.2022.102956
• 发表时间: 2022
• 期刊: Parallel Computing
• 影响因子: 1.4
• 作者: Xu, Tianshi;Kalantzis, Vassilis;Li, Ruipeng;Xi, Yuanzhe;Dillon, Geoffrey;Saad, Yousef
• 通讯作者: Saad, Yousef

Proxy-GMRES: Preconditioning via GMRES in Polynomial Space

Proxy-GMRES：通过多项式空间中的 GMRES 进行预处理

• DOI: 10.1137/20m1342562
• 发表时间: 2021
• 期刊: SIAM Journal on Matrix Analysis and Applications
• 影响因子: 1.5
• 作者: Ye, Xin;Xi, Yuanzhe;Saad, Yousef
• 通讯作者: Saad, Yousef

Data-Driven Construction of Hierarchical Matrices With Nested Bases

具有嵌套基的分层矩阵的数据驱动构建

• DOI: 10.1137/22m1500848
• 发表时间: 2023
• 期刊: SIAM Journal on Scientific Computing
• 影响因子: 3.1
• 作者: Cai, Difeng;Huang, Hua;Chow, Edmond;Xi, Yuanzhe
• 通讯作者: Xi, Yuanzhe