Tensor Models, Methods, and Medicine

目前对研究大规模数据的有效、定量和可解释的方法的需求是前所未有的。通常的情况是，该数据自然是多峰的，并且由张量很好地表示，张量是可以由多维阵列表示的公共矩阵的高阶推广。因此，人们对张量的数学产生了浓厚的兴趣，但由于这一领域的问题往往比矩阵的类似问题复杂得多，因此在基于张量的数据分析技术的开发过程中存在着关键的空白，特别是在寻求自动学习复杂数据集的潜在趋势或主题的主题建模领域。事实上，实践者在应用基于矩阵的主题建模技术之前，通常必须执行昂贵的张量数据到矩阵的转换，该技术无法从被丢弃的模式中检测到数据中的潜在信息；这种信息损失在医学成像等敏感应用中尤其危险。这个项目试图填补这些空白，并为张量主题建模提供工具，以自然的形式处理数据。该团队将与港湾-加州大学洛杉矶分校医学中心心脏病学系的合作者合作，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具有多模式数据的领域的实用模型，以及促进对这些模型、它们的训练方法以及它们所应用的复杂张量数据的理论理解。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式结构，允许纳入灵活的监督信息，并识别分层主题结构。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，为关键子例程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，第三个目标是说明这些模型和方法在超声心动图分析的重要案例研究中的应用前景。综合起来，这些目标将拓宽张量分析的数学基础，同时也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的知识，限制对即使是张量专家也不太了解的模型参数的依赖。除了这种社会影响，该项目还包括一个强大的外展和教育部分，将为本科生参与者提供形成研究的机会，并促进应用领域专家和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，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.