III: Small: Large-Scale Structured Sparse Learning

最近的技术革命导致数据的规模、多样性和复杂性急剧增长。现代数据分析在处理这种复杂性方面面临着新的挑战。尽管复杂，但许多现实世界数据的底层表示通常是稀疏的。这种稀疏性通常表现出内在结构，例如，空间或时间平滑、图、树和组。发现有效的稀疏表示对于科学发现是至关重要的，先验结构信息可以显著改进稀疏学习模型。该项目正在开发能够从海量高维和复杂数据中发现知识的算法和工具(包括开源软件)，以及将所提出的研究纳入课堂的新课程。大多数稀疏学习算法由于其稀疏性和强大的理论保证而基于L1范数，但这并不能捕获结构。该项目通过(1)分析与各种特征结构相关联的所谓邻近算子，解释它们如何以及为什么能够产生期望的结构化稀疏性；(2)开发高效的计算邻近算子的算法，它在所提出的优化算法中起到关键的构建块作用；(3)开发一个结构化稀疏学习框架，其中包括本项目中开发的各种稀疏学习模型和算法。

Recent technological revolutions have lead to dramatically growing scale, diversity, and complexity of data. Modern data analysis is facing new challenges in handling this complexity. Although complex, the underlying representations of many real-world data are often sparse. This sparseness often exhibits intrinsic structure, e.g., spatial or temporal smoothness, graphs, trees, and groups. Finding effective sparse representations is fundamentally important for scientific discovery; the a-priori structure information may significantly improve the sparse learning model. This project is developing algorithms and tools (including open source software) to enable knowledge discovery from massive high-dimensional and complex data, as well as a new curriculum that incorporates the proposed research into the classroom.Most sparse learning algorithms are based on the L1 norm due to its sparsity-inducing property and strong theoretical guarantees, but this does not capture structure. This project is advancing structured sparse learning by (1) analyzing the so-called proximal operators associated with various feature structures, which explains how and why they can induce the desired structured sparsity; (2) developing efficient algorithms for computing the proximal operators, which plays a key building block role in the proposed optimization algorithms; (3) developing a structured sparse learning framework, which includes various sparse learning models and algorithms developed in this project.