Multi-aspect data arise ubiquitously in sensing, communications, and signal processing. For example, a hyperspectral image in remote sensing has two spatial aspects and one spectral aspect; spectrum dynamics data in wireless communications comprise frequency, time, and space aspects. As higher-order extensions of matrices, tensor models are considered indispensable tools for multi-aspect sensing and analytics. This project is concerned with a recently emerged tensor decomposition model, namely, the block-term decomposition model in multilinear rank-(L,L,1) terms (i.e., the LL1 model). This model is well-aligned to the underlying physics of a number of core applications in remote sensing, medical imaging, chemometrics, and wireless communications, thereby offering strong theoretical guarantees for these tasks. However, the LL1 model is relatively new, and thus many challenges pertaining to its theory and methods (e.g., scalability, robustness, and missing value recoverability) are still largely uncharted territories. The appealing promises for boosting performance of engineering applications have been barely fleshed out. This project will significantly advance the understanding to the computational and analytical aspects of the LL1 model --- leading to a series of theory-backed refreshing multi-aspect data acquisition and processing algorithms. Beyond engineering, the developed theory and methods will also be broadly applicable in related domains such as ecology, biology and food science, where multi-aspect data frequently come up. The project will also offer opportunities for training undergraduate students in optimization, linear algebra, and real-data acquisition.
Towards fully capitalizing the power of the LL1 tensor model, this project will address a number of critical challenges in LL1 decomposition theory and methods. Specifically, the first thrust will design scalable LL1 algorithms that can flexibly incorporate a large variety of prior information as constraints and regularization; the second thrust will develop model identification guarantees and algorithms for LL1 computations in the presence of gross outliers; the third thrust will develop theory and algorithms for recovering compressed/downsampled LL1 tensors; and the last thrust applies the proposed approaches onto real-world engineering problems such as hyperspectral unmixing, fluorescence data analysis, hyperspectral super-resolution, and spectrum cartography. The proposed computational and analytical tools are well-motivated in wake of the rapid growth of multi-aspect data. Optimization problems associated with LL1 tensor decomposition are hard in both theory and practice, due to their nonconvex nonsmooth nature and the LL1 model’s inherent ill-conditioned multilinear structure. Outlier-robust LL1 decomposition and compressed/downsampled LL1 tensor recovery pose exciting research questions that reside at the core of signal processing and data analytics. The solutions developed in this project will offer new insights and effective tools for these challenging open problems, which are bound to benefit a broad range of real-world applications.
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.
