AF:Small: Divide-and-Conquer Numerical Methods for Analysis of Massive Data Sets

在医疗保健、基因组学、能源管理和社会网络分析等各种应用中，数据正在以惊人的速度产生。事实上，最近的大数据（Big Data）这个名字强调了海量数据无处不在。因此，非常需要开发可扩展的复杂方法来分析这些数据集。该项目旨在解决这一挑战的一个方面，即为机器学习中出现的现代问题开发可扩展的最先进的数值方法。该项目旨在为当代应用中出现的具有代表性的具体问题开发分而治之的方法。这些包括(a)分类：核支持向量机；(b)回归：核回归和高维稀疏逼近；(c)结构学习：图模型估计；(d)谱近似：多尺度SVD计算；(e)缺失值估计：矩阵分解。该项目将为这些问题中的每一个开发专门的算法，特别是开发将问题划分为子问题，解决子问题并最终征服子问题的定制方法。因此，将分而治之的方法应用于大规模机器学习中的其他问题的一般原则将被揭示。该项目将开发出在现代多核计算机上高效运行的大规模数据分析软件。将研究新算法在生物信息学和网络分析等各个应用领域的影响。在计算机科学和应用数学领域，该项目将对各种学科的研究产生广泛的影响，包括数值分析、数值优化、统计学、机器学习、数据挖掘和并行计算。

Data is being generated at a tremendous rate in diverse applications, such as health care, genomics, energy management and social network analysis. Indeed, the recent moniker of Big Data emphasizes that massive volumes of data are ubiquitous. Thus, there is a great need for developing scalable and sophisticated methods for analyzing these data sets. This project is aimed towards one aspect of this challenge, namely, developing scalable and state-of-the-art numerical methods for modern problems that arise in machine learning. This project will aim to develop divide-and-conquer methods for representative, concrete problems that arise in contemporary applications. These include (a) classification: kernel support vector machines, (b) regression: kernel regression and high-dimensional sparse approximation, (c) structure learning: graphical model estimation, (d) spectral approximation: multi-scale SVD computation, and (e) missing value estimation: matrix factorization. The project will develop specialized algorithms for each of these problems, in particular, developing tailored ways of dividing the problem into subproblems, solving the subproblems, and finally conquering the subproblems. Thus, general principles for applying the divide-and-conquer approach to other problems in large-scale machine learning will be uncovered. The project will lead to software for large-scale data analysis that will be efficient on modern multi-core computers. Impact of the new algorithms on various application areas, such as bioinformatics and network analysis, will be studied. Within computer science and applied mathematics, the project will have a broad impact on research in a variety of disciplines, including numerical analysis, numerical optimization, statistics, machine learning, data mining and parallel computing.