Tensor Models, Methods, and Medicine

目前，人们对高效、定量和可解释的方法来研究大规模数据的需求是前所未有的。通常情况下，这些数据自然是多模态的，并且可以用张量很好地表示，张量是可以用多维数组表示的公共矩阵的高阶推广。由于这个原因，人们对张量数学的兴趣激增，但由于这一领域的问题往往比矩阵的类似问题复杂得多，因此在转化为基于张量的数据分析技术的发展方面存在关键差距，特别是在主题建模领域，该领域寻求自动学习复杂数据集的潜在趋势或主题。事实上，从业者通常必须在应用基于矩阵的主题建模技术之前将其张量数据转换为矩阵，该技术无法从丢弃的模式中检测到数据中的潜在信息;这种信息丢失在敏感应用中特别危险，如医学成像。 该项目旨在填补这些空白，并提供用于张量主题建模的工具，以自然形式处理数据。该团队将与Harbor-UCLA Medical Center Department of Cardiology的合作者合作，将这些工具应用于心脏成像数据的案例研究，提供直接的社会影响，并指导数学技术的发展。该项目将提供可应用于任何领域的多模态数据的实用模型，并促进对这些模型的理论理解，其训练方法，以及它们所应用到的复张量数据。PI侧重于三个主要目标。第一个目标是开发基于张量的主题模型，该模型尊重数据的自然多模态结构，允许纳入灵活的监督信息，并确定分层主题结构。第二个目标是为基于张量的主题模型设计高效，低内存和在线训练方法，为关键子程序提供收敛保证和复杂性分析，并产生公开可用的开源实现。最后，第三个目的是说明这些模型和方法的承诺，在一个重要的案例研究应用超声心动图分析。这些目标将拓宽张量分析的数学基础，同时将张量分析的应用扩展到重要和敏感的现实世界领域。所开发的模型和方法将产生广泛的影响，因为它们可以利用领域专家知识，并限制对模型参数的依赖，即使是张量专家也不能很好地理解。除了这种社会影响，该项目还包括一个强大的推广和教育部分，将为本科参与者提供形成性研究的机会，并促进应用领域专家与数学和数据科学技术专家之间的合作。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

There is currently an unprecedented demand for efficient, quantitative, and interpretable methods to study large-scale data. It is often the case that this data is naturally multi-modal and represented well by a tensor, a higher-order generalization of the common matrix which can be represented by a multi-dimensional array. For this reason, there has been a surge of interest in the mathematics of tensors, but as questions in this area are often far more complex than the analogous questions for matrices, there are key gaps in translation to development of tensor-based data analytic techniques, especially in the area of topic modeling, which seeks to automatically learn latent trends or topics of complex data sets. Indeed, practitioners often must perform a costly transformation of their tensor data into a matrix before applying matrix-based topic modeling techniques that fail to detect latent information in the data from the discarded modes; such loss of information is especially dangerous in sensitive applications like medical imaging. This project seeks to fill these gaps and to provide tools for tensor topic modeling that treat the data in its natural form. The team will partner with collaborators in the Harbor-UCLA Medical Center Department of Cardiology to apply these tools to case study cardiac imaging data, providing direct societal impact as well as directing the development of the mathematical techniques.This project will provide practical models that can be applied in any field with multi-modal data, as well as to advance the theoretical understanding of these models, their training methods, and the complex tensor data to which they are applied. The PI focuses on three main aims. The first aim is to develop tensor-based topic models which respect the natural multi-modal structure of the data, allow for incorporation of flexible supervision information, and identify hierarchical topic structure. The second aim is to design efficient, low-memory, and online training methods for tensor-based topic models, provide convergence guarantees and complexity analysis for key subroutines, and produce publicly available open-source implementations. Finally, the third aim is to illustrate the promise of these models and methods in an important case study application to echocardiogram analysis. Together, these aims will broaden the mathematical foundations of tensor analysis while also expanding application of tensor analysis to important and sensitive real-world domains. The developed models and methods will have wide impact as they can utilize domain expert knowledge and limit dependence on model parameters that even tensor experts do not understand well. In addition to this societal impact, the project includes a strong outreach and educational component that will provide formative research opportunities to undergraduate participants, and promote collaboration between application domain experts and experts in mathematical and data-scientific techniques.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.