BIGDATA: Collaborative Research: F: Foundations of Nonconvex Problems in BigData Science and Engineering: Models, Algorithms, and Analysis

在当今的数字世界中，几乎在科学研究和人类活动的各个方面都可以找到大量的数据，即大数据。需要对这些数据进行有效管理，以进行可靠的预测和推断，从而改进决策。统计学习是一门新兴的科学学科，其中数学建模、计算算法和统计分析被联合使用来解决这些具有挑战性的数据管理问题。不可避免地，需要为整个学习过程引入定量标准，以便衡量所获得的解决方案的质量。本研究主要关注两个重要的标准：数据适应度和基础学习模型的稀疏性表示。这些结果的潜在应用可以在计算统计学、压缩感知、成像、机器学习、生物信息学、投资组合选择和不确定决策等许多涉及大数据的领域中找到。到目前为止，凸优化一直是统计学习的主要方法，其中所采用的两个标准由凸函数表示，要么被优化，要么被设置为所寻找变量的约束。最近，凸差分（DC）类型的非凸函数和凸差分算法（DCA）已被证明在许多情况下产生优越的结果，这是本项目的动机。目标是建立一个坚实的基础和统一的框架，以解决大数据问题中的许多基本问题，其中所要解决的优化问题存在非凸性和不可微性。在计算统计学习中，这两个非标准的特征是具有挑战性的，它们的严格处理需要融合来自不同数学科学领域的专业知识。要研究的技术问题将涵盖统计学习及其许多应用中产生的非凸、非光滑优化问题的可计算解的最优性、稀疏性和统计特性。新的算法将首先在用于初步实验的合成数据集上开发和测试，然后在用于现实的公开可用数据集上测试；将对学习问题的不同表述进行比较。

In today's digital world, huge amounts of data, i.e., big data, can be found in almost every aspect of scientific research and human activity. These data need to be managed effectively for reliable prediction and inference to improve decision making. Statistical learning is an emergent scientific discipline wherein mathematical modeling, computational algorithms, and statistical analysis are jointly employed to address these challenging data management problems. Invariably, quantitative criteria need to be introduced for the overall learning process in order to gauge the quality of the solutions obtained. This research focuses on two important criteria: data fitness and sparsity representation of the underlying learning model. Potential applications of the results can be found in computational statistics, compressed sensing, imaging, machine learning, bio-informatics, portfolio selection, and decision making under uncertainty, among many areas involving big data.Till now, convex optimization has been the dominant methodology for statistical learning in which the two criteria employed are expressed by convex functions either to be optimized and/or set as constraints of the variables being sought. Recently, non-convex functions of the difference-of-convex (DC) type and the difference-of-convex algorithm (DCA) have been shown to yield superior results in many contexts and serve as the motivation for this project. The goal is to develop a solid foundation and a unified framework to address many fundamental issues in big data problems in which non-convexity and non-differentiability are present in the optimization problems to be solved. These two non-standard features in computational statistical learning are challenging and their rigorous treatment requires the fusion of expertise from different domains of mathematical sciences. Technical issues to be investigated will cover the optimality, sparsity, and statistical properties of computable solutions to the non-convex, non-smooth optimization problems arising from statistical learning and its many applications. Novel algorithms will be developed and tested first on synthetic data sets for preliminary experimentation and then on publicly available data sets for realism; comparisons will be made among different formulations of the learning problems.