BIGDATA: F: DKA: CSD: DKM: Theory and Algorithms for Processing Data with Sparse and Multilinear Structure

张量，或多维数组结构，自然出现在大数据应用中，如心理测量学、多媒体、社交媒体、基因组学、神经成像、地理空间数据和湍流模拟。这些应用程序中的张量具有大量数据，并且难以存储、传输、计算和分析。为了成功地执行这些任务，发现和利用数据中的相关和信息结构是至关重要的，在张量上下文中，这些结构通常具有分解或分解为相对少量的简单成分的形式。不幸的是，与更简单的矩阵情况相比，现有的张量数学理论和计算工具受到了严重的限制，不能满足大数据应用的需要。该项目的总体目标是缩小大数据张量工具的理论和实践差距。由于张量在大数据中的基本和无处不在的作用，本研究的结果有可能影响大数据感兴趣的每个领域。特别是，新的工具和方法有可能使神经科学领域的新的基础发现成为可能，对人类健康有重要的好处。更具体地说，本研究旨在研究具有理论性能保证的计算效率高的算法。这项工作强调高度可扩展的在线和分布式版本，接近基本极限。方法包括松弛实现低秩分解，确定压缩感知的基本限制，以及考虑实际问题，如噪声数据。这些算法在多模态功能神经成像和神经科学的大数据应用中得到了验证。由于它吸收并包括数学、计算机科学、工程学、统计学和神经科学，这项研究是高度多学科的。

Tensors, or multidimensional array structures, naturally arise in big data applications such as psychometrics, multimedia, social media, genomics, neuroimaging, geospatial data, and turbulent flow simulations. Tensors in these applications have massive amounts of data and are difficult to store, transmit, compute with, and analyze. To perform these tasks successfully it is crucial to discover and utilize relevant and informative structure within the data, which, in the tensor context, often has the form of a decomposition or factorization into a relatively small number of simpler constituents. Unfortunately, compared to the simpler matrix case, existing mathematical theory and computational tools for tensors are severely limited, and do not meet the needs of big data applications. The broad goal of this project is to close this theoretical and practical gap in tools for tensors for big data. Because of the fundamental and ubiquitous role of tensors in Big Data, the results of this research have the potential to impact every field in which big data is of interest. In particular, the new tools and methodology have the potential to enable new fundamental discoveries in neuroscience, with important benefits to human health.More specifically, this research aims for computationally efficient algorithms with theoretical performance guarantees. The work emphasizes highly scalable online and distributed versions, approaching the fundamental limits. Approaches include relaxations to achieve low rank decomposition, identifying fundamental limits for compressed sensing, and consideration of practical issues such as noisy data. These algorithms are validated on big data applications in multimodality functional neuroimaging and neuroscience. As it draws on and includes mathematics, computer science, engineering, statistics, and neuroscience, this research is highly multidisciplinary.