Collaborative Research: Sparse Optimization in Large Scale Data Processing: A Multiscale Proximity Approach

协作研究：大规模数据处理中的稀疏优化：多尺度邻近方法

• 批准号: 1912958
• 负责人: Yuesheng Xu
• 金额: $ 12.5万
• 依托单位: Old Dominion University Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1912958&HistoricalAwards=false
• 关键词: Collaborative; Research; Sparse; Optimization; Large

There is an emergent demand in areas of national strategic interest such as information technology, nanotechnology, biotechnology, civil infrastructure and environment for abstracting useful knowledge for decision making or uncovering truth from large-scale data acquired via various means such as sensors and internet. A core issue of these areas is to develop accurate mathematical models, which govern the abstraction process, and to design efficient algorithms that solve the underlying optimization problems for the models. A challenge of the tasks comes from the large-scale nature of given data. This nature requires determining a large number of model parameters and it is computationally expensive. To address this challenge, this project will take advantage of certain intrinsic multiscale structure of given data in modeling so that the resulting models have significantly fewer parameters to be determined. It is also crucial to introduce efficient algorithms for solving the resulting optimization problems for the models, which have intrinsic multiscale structures. The second goal of this proposed research is to provide rigorous training of young mathematicians and computational scientists so that they have the skill sets needed to face the challenges of the big data era through this proposed research and its associated educational components. Outcomes of the proposed research and its educational component will certainly contribute to the Federal strategic interest areas.This research project addresses several critical issues of processing large-scale data, such as high dimensionality and high noise, through properly choosing structured sparsity promoting non-convex functions in modeling and through synthesizing the multiscale representation of data and using fixed-point equations/inclusions involved the proximity operator in solving the resulting optimization problem. Structured non-convex sparsity promoting functions are proposed to overcome drawbacks of the existing modeling of large-scale data, leading to the design of efficient single-scale proximity algorithms. Multiscale analysis has been developed to efficiently represent data, while how multiscale representation of data is used to improve convergence of the fixed-point proximity algorithm remains unsolved. The proposed multiscale proximity method avoids iterations on the full large-scale of the fixed-point equation/inclusion. Instead, when data are represented in a multiscale analysis, iterations of the multiscale proximity algorithm are conducted only on a (small-scale) lower frequency component of the equation/inclusion (based on a single-scale algorithm), and only one functional evaluation on a (large-scale) high frequency component is required. The multiscale algorithm will preserve accuracy of the single-scale algorithm while accelerating its convergence significantly. This leads to a fast algorithm for solving the fixed-point equation/inclusion involved the proximity operator.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.

信息技术、纳米技术、生物技术、民用基础设施和环境等国家战略利益领域迫切需要从传感器和互联网等各种手段获取的大规模数据中提取有用的知识以供决策或揭示真相。这些领域的核心问题是开发精确的数学模型来控制抽象过程，并设计有效的算法来解决模型的底层优化问题。这些任务的挑战来自给定数据的大规模性质。这种性质需要确定大量的模型参数，并且计算成本很高。为了应对这一挑战，该项目将在建模中利用给定数据的某些内在多尺度结构，从而使生成的模型需要确定的参数显着减少。引入有效的算法来解决具有内在多尺度结构的模型的优化问题也至关重要。这项拟议研究的第二个目标是为年轻数学家和计算科学家提供严格的培训，使他们拥有通过这项拟议研究及其相关教育部分应对大数据时代挑战所需的技能。拟议研究的成果及其教育部分肯定将为联邦战略利益领域做出贡献。该研究项目通过正确选择促进非凸的结构化稀疏性，解决了处理大规模数据的几个关键问题，例如高维和高噪声建模中的函数以及通过综合数据的多尺度表示以及使用定点方程/包含物涉及邻近算子来解决所产生的优化问题。提出了结构化非凸稀疏性提升函数来克服现有大规模数据建模的缺点，从而设计出高效的单尺度邻近算法。多尺度分析已经被开发来有效地表示数据，而如何使用数据的多尺度表示来提高定点邻近算法的收敛性仍然悬而未决。所提出的多尺度邻近方法避免了定点方程/包含物的整个大规模上的迭代。相反，当数据在多尺度分析中表示时，多尺度邻近算法的迭代仅在方程/包含物的（小尺度）低频分量（基于单尺度算法）上进行，并且仅进行一次功能评估需要（大规模）高频分量。多尺度算法将保持单尺度算法的准确性，同时显着加速其收敛。这导致了一种用于求解涉及邻近算子的定点方程/包含式的快速算法。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力优点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

Regularization in a functional reproducing kernel Hilbert space

• DOI: 10.1016/j.jco.2021.101567
• 发表时间: 2021-03
• 期刊: J. Complex.
• 作者: Rui Wang;Yuesheng Xu

Sparse Deep Neural Network for Nonlinear Partial Differential Equations

非线性偏微分方程的稀疏深度神经网络

• DOI: 10.4208/nmtma.oa-2022-0104
• 发表时间: 2022
• 期刊: Methods and Applications
• 作者: null, Yuesheng Xu;Zeng, Taishan
• 通讯作者: Zeng, Taishan

A content-adaptive unstructured grid based regularized CT reconstruction method with a SART-type preconditioned fixed-point proximity algorithm

• DOI: 10.1088/1361-6420/ac490f
• 发表时间: 2022-03-01
• 期刊: INVERSE PROBLEMS
• 影响因子: 2.1
• 作者: Chen，Yun;Lu，Yao;Xu，Yuesheng
• 通讯作者: Xu，Yuesheng

Deeply learning deep inelastic scattering kinematics

深度学习深度非弹性散射运动学

• DOI: 10.1140/epjc/s10052-022-10964-z
• 发表时间: 2022
• 期刊: The European Physical Journal C
• 作者: Diefenthaler, Markus;Farhat, Abdullah;Verbytskyi, Andrii;Xu, Yuesheng
• 通讯作者: Xu, Yuesheng

Representer Theorems in Banach Spaces: Minimum Norm Interpolation, Regularized Learning and Semi-Discrete Inverse Problems

• 发表时间: 2020-06
• 期刊: J. Mach. Learn. Res.
• 作者: Rui Wang;Yuesheng Xu