Optimization problems with low-dimensional tensor structure

In this thesis, we consider optimization problems that involve statistically estimating signals from tensor data. We observe that modelling the signals as tensors using tensor factorizations increases the accuracy of more closely estimating the true signal. We empirically show that this is a result of two reasons: (i) it reduces the number of parameters that need to be estimated, hinting at a decrease in sample complexity, and (ii) it allows us to take advantage of the low-rank property that many high-dimensional tensor data samples possess. We also show that while there exists a tradeoff between the accuracy of the reconstructed signal and the ranks according to the tensor decomposition, there often exists a rank that improves performance in sample-starved settings. We discuss these tradeoffs and develop algorithms for two of these applications, classification and phase retrieval, and demonstrate the effectiveness of our algorithms under several different settings and performance metrics.M.S.Includes bibliographical reference