III: Medium: High-Performance Factorization Tools for Constrained and Hidden Tensor Models

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Tensors generalize matrices to higher dimensions (called modes) and are designed to model multi-way data. Tensor factorization algorithms analyze such multi-way data to uncover relations between the different modes that can be used to both gain insights and to predict unknown aspects of the underlying system/process. For example, medical diagnosis and treatment records can be modeled via a four-mode tensor whose modes correspond to patients, physicians, diagnosis, and treatments and its factorization can provide insights on the co-occurrence of medical conditions, treatment approaches, any treatment differences based on the physician, and identify potential instances of medical fraud. This project's research is designed to address current limitations of tensor analysis by developing new theory and algorithms and high-performance scalable parallel formulations of the various computational kernels used by these algorithms, and a flexible open source software toolkit that can be used to perform constrained and hidden tensor factorization of very large and sparse multi-way datasets. The success of this project will allow researchers to leverage the power of multi-way ``Big Data'' analysis to solve various problems in diverse application domains such as healthcare, medical imaging, cybersecurity, social and behavioral sciences, and e-commerce. At the same time, the project will provide data science training to the students involved by combining cutting-edge data and signal analytics, data mining, and high-performance computing.Constrained matrix and tensor factorization techniques are widely used for dimensionality reduction, clustering, and estimation in machine learning, signal processing, and many other walks of science and engineering. Unconstrained matrix and tensor factorization algorithms are relatively mature, but constrained counterparts are lagging in terms of speed, scalability, and flexibility. In many applications (e.g., medical imaging and recommender systems), instead of observing the actual entries of a tensor, we observe a limited number of linear combinations (e.g., partial sums) of these entries and need to identify the tensor's latent factors from these measurements. Being able to directly identify the latent factors from linear measurements, which we refer to as hidden tensor factorization, has important advantages in terms of complexity, memory footprint, and the ability to handle very large data sets. Developing open source high-performance parallel tools for constrained and hidden tensor factorization in both shared- and distributed-memory systems will significantly enhance the ability to analyze very large multi-way data. The research will evolve along two synergistic thrusts. First, it will develop new theory and algorithms for constrained and hidden tensor factorization by (i) building fast first-order (FFO) and fast stochastic first-order (FSFO) constrained tensor decomposition algorithms that strike favorable trade-offs between simplicity, scalability, and speed of convergence, and (ii) tackling important identifiability and algorithmic issues related to hidden tensor factorization. Second, it will undertake a multi-pronged effort towards developing high-performance parallel formulations for the computational kernels used in constrained and unconstrained tensor and hidden tensor factorization and develop a high-performance tensor factorization software toolbox. The release of the high-performance tensor factorization toolbox will enable researchers and practitioners to scale up not only the size of data but also the variety of constraints and types of data they can analyze. The research will involve students that will be trained in data science, combining cutting-edge signal and data analytics, data mining, and high-performance computing.