Computational techniques for analysis of mixed signals

Mixtures of signals appear in a variety of scientific and industrial domains and are often analyzed with state-of-the-art deep learning techniques. This approach is well suited to applications with plentiful labelled data such as imaging, but requires too many training examples areas where large scale data collection is impractical or expensive. This dissertation explores practical methods and surrounding computational tools to analyze and separate mixtures. We explore modeling these signal demixing problems as tensor factorizations, engineer a coding package in the Julia language to factorize these tensors, and study new optimization techniques for performing this efficiently. These techniques include matrix and tensor rank estimation by curvature maximization, accelerated descent with momentum and Hessian approximation, rescaling to enforce constraints rather than Euclidean projections, and a multiscale optimization framework. Importantly, these advancements reach beyond signal demixing and into the fields of numerical analysis, operations research, and computer science as a whole. These efforts would remain theoretical without the explicit examples provided to test these developments on real-world applications in the fields of sediment analysis, spatial transcriptomics, and stem separation. Finally, we lay the groundwork for future study and open problems related to this problem.