Computation of Large-Scale, Multi-Dimensional Sparse Optimization Problems

所提出的研究主要在于稀疏优化，这是从密集数据集中发现稀疏或其他结构简单的解的优化研究的一个新的独特领域。它的发展借鉴了经典非线性规划的算法技术，并受到数据科学许多其他领域的发展的推动。如今，实例的大小、复杂性和多样性都有了显著增长。提出的研究从以下几个方面解决了这些新的挑战：处理多正则化子的数据和变量拆分，并行和分布式优化，有效的模型路径计算和正则化参数选择，随机逼近，以及用于非凸优化的坐标下降方法。这些研究有望显著减少现有算法的运行时间，产生新的算法，使之能够解决目前在数据科学中无法解决的广泛问题。特别是，预期的结果将使机器学习模型适合以前无法访问的数据(例如，分布式数据)，允许在更高维度和跨不同模式的数据挖掘，以及以可计算处理的方式处理多个正则化程序。数据收集技术的进步导致了互联网、工程、气候研究、宇宙学和医学等不同领域的大数据的快速扩散。为了让这些海量数据有意义，科学家和工程师正在引入新的计算方法来分析他们的数据。在这些方法中，稀疏优化和结构化解决方案变得非常重要。如今，他们的业务范围正在迅速扩大。除了一维信号和二维图像的传感和处理外，三维视频、四维CT和多向张量等高维量已经成为模型中的数据或未知变量。除了稀疏结构之外，诸如低秩度、稀疏图、树结构、几个字典原子的线性表示及其组合等结构已经在包括基因组图谱、蛋白质结构研究、社会网络分析、股票价格预测和文本/语音挖掘的各种应用中以期望的结构首次出现。拟议的研究将建立在最近的成功基础上，并导致处理大型、不同类型的数据和变量的新技术、寻求解的各种结构的新算法、将现有的数值方法扩展到并行和分散的计算体系结构，以及对解决上述几个应用领域的关键问题的贡献。

The proposed research largely lies in sparse optimization, a new distinct area of research in optimization for discovering sparse or other simple-structured solutions from dense datasets. Its development draws algorithmic techniques from classical nonlinear programming and is nurtured by the development in many other areas of data science. Today, the size, complexity, and diversity of instances have grown significantly. The proposed research addresses these new challenges in the following directions: data and variable splitting for handling multiple regularizers and for parallel and distributed optimization, efficient model path computation and regularization parameter selection, stochastic approximation, and coordinate descent methods for non-convex optimization. These investigations are expected to significantly reduce the running times of the existing algorithms, giving rise to novel algorithms to enable the solutions of a wide ranges of problems that are currently not solvable in data sciences. In particular, the expected results will fit machine learning models to data previously inaccessible (e.g., distributed data), enable the mining of data in much higher dimensions and across different modalities, as well as handle multiple regularizers in a computationally tractable way.Technological advances in data gathering have led to a rapid proliferation of big data in diverse areas such as the Internet, engineering, climate studies, cosmology, and medicine. In order for this massive amount of data to make sense, new computational approaches are being introduced to let scientists and engineers analyze their data. Among these approaches, sparse optimization and structured solutions have grown enormously important. Today, their scopes are quickly expanding. Beyond the sensing and processing of 1D signals and 2D images, high-dimensional quantities such as 3D video, 4D CT, and multi-way tensors have become the data or unknown variables in models. Beyond the sparsity structure, structures such as low-rankness, sparse graph, tree structure, linear representation of a few dictionary atoms, as well as their combinations, have debut as desired structures in various applications including genome mapping, protein structure study, social network analysis, stock price prediction, and text/speech mining. The proposed research will build on the recent successes and lead to new techniques for handling large-sized, diverse-typed data and variables, novel algorithms for pursuing a variety of structures in solutions, the extension of existing numerical methods to parallel and decentralized computing architectures, and the contributions to solving key problems in several aforementioned application areas.