Tensor Models, Methods, and Medicine

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.