Jamie Haddock University of California, Los Angeles (UCLA) Mathematics Datasets with hierarchical structure arise in a wide variety of fields, such as document classification, image processing, and bioinformatics. A fundamental task in these areas is to identify the latent hierarchical topic structure in data. My work in mathematical data science spans both modeling and algorithmic methods to attack such problems. In this talk, I will describe first a model for detecting latent hierarchical topic structure based upon nonnegative matrix factorization. We show this model can be endowed with a feed-forward neural network structure, allowing it to be trained via forward and back propagation, a scheme we call Neural NMF. Second, I will discuss the family of algorithms I have developed which solve a fundamental subroutine in Neural NMF. My analysis of these algorithms use insights from probability theory, convex geometry, and numerical analysis. Finally, I will demonstrate some empirical advantages Neural NMF has over typical methods and apply it to MyLymeData.