Collaborative Research: Sparse Optimization for Machine Learning and Image/Signal Processing

涉及信息技术、纳米技术、生物技术、民用基础设施、环境科学和其他重要领域的数据集通常非常庞大。由于数据量和相关模型大小的不断增长，对更有能力的数据处理模型的需求不断增加。应用程序中的数据集通常具有某些嵌入的稀疏结构，从某种意义上说，它们的基本内在特征可以由更少量的信息表示。受此观察的启发，该项目的目的是通过探索嵌入在数据集中的稀疏结构，为非光滑和非凸优化开发计算效率高的方法。研究人员预计，该项目的成果将在许多应用领域中使用。该项目中的研究和相关教育部分预计将为本科生和研究生提供严格的培训，使他们具备应对大数据时代科学和技术挑战所需的技能。 该项目将解决非光滑、非凸优化中的几个关键问题，这些问题来自于一系列应用中的数据稀疏建模，包括机器学习和稀疏图像/信号恢复。对于学习和图像/信号恢复，将开发适当的稀疏正则化模型。将设计合适的稀疏性促进函数用于形成正则化项，并构造适当的基/变换，使得所得到的正则化算法迫使它们的解是稀疏的。对于机器学习，将建立适当的再生核Banach空间，以允许表示数据的复杂和丰富的几何和拓扑结构，从而提高学习效果。将建立这些空间中的学习方法的表示定理，使得学习解决方案可以表示为有限数量的内核会话的组合，其数量等于训练中使用的数据点的数量，即使假设空间是无限维的。这些空间的几何特征将被用来诱导稀疏的学习解决方案。通过对数据的机器学习构建基础或转换，有望改进图像/信号恢复方法。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。

Data sets involved in information technology, nanotechnology, biotechnology, civil infrastructure, environmental science, and other important areas are often extremely large. Due to growing quantities of data and related model sizes, demands for more competent data processing models continue to increase. Data sets in applications often have certain embedded sparsity structures, in the sense that their essential intrinsic characteristics can be represented by smaller amounts of information. Motivated by this observation, the aim of this project is to develop computationally efficient methods for non-smooth and non-convex optimization by exploring sparsity structures embedded in data sets. The investigators anticipate that the outcomes of this project will be of use in many application areas. The research and associated educational components in this project are expected to provide undergraduate and graduate students rigorous training so that they will have the skill sets needed to face the scientific and technological challenges of the big data era. This project will address several critical issues in non-smooth, non-convex optimization that result from sparse modeling of data in a range of applications, including machine learning and sparse image/signal restoration. For both learning and image/signal restoration, proper sparse regularization models will be developed. Suitable sparsity promoting functions will be designed for use in forming regularization terms, and appropriate bases/transforms will be constructed so that the resulting regularization algorithms compel their solutions to be sparse. For machine learning, appropriate reproducing kernel Banach spaces will be built up to allow for the representation of complex and rich geometric and topological structures of data and hence lead to improved learning outcomes. Representer theorems of the resulting learning methods in these spaces will be established so that the learning solutions can be expressed as a combination of a finite number of kernel sessions with the number equal to that of the data points used in training even though the hypothesis space is of infinite dimension. Geometric features of these spaces will be employed to induce sparsity of the learning solutions. The construction of bases or transforms via machine learning from data is expected to lead to improved methods for image/signal restoration.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.