Gradient Methods for Solving Big Data (Tensor) Optimization Problems

Rapid developments in modern technologies have made large-scale statistical data sets readily available in many industrial settings, but the task of effectively turning data into useful information still remains a major challenge in a wide range of applications. One important source of the 'big data' complication stems from the way in which the data points are collected and stored. In particular, the data format in question is known as the tensor, which is a useful format, because it reflects the interconnections between various factors. Tensor data sets can be found in statistical learning, bioinformatics, consumer behavior in marketing, climate change studies, and signal and image processing. However, computationally the tensor data formats are notoriously difficult. This award supports fundamental research on the computational aspects of the above-mentioned information retrieval process. The project has a multidisciplinary research element and will positively impact engineering education.

In the context of tensor data processing, a desirable operation is to compute the projection of a given data tensor onto a simpler set with a certain low complexity structure, where the 'low complexity' tensors may refer to low-rank tensors, or sparse tensors, or it may also refer to the tensors with low co-cluster numbers. Formulating tensor projection and completion problems leads to large scale non-convex -- yet algebraic -- optimization models. This research will develop a framework for the iteration complexity analysis which will enable effective first-order computational methods for tensor projection, completion and optimization models.