UWEC CERCA 2026

Every deep learning model builds an internal representation of its training data in a high-dimensional geometric object called a latent or embedding space whose shape encodes what the model has learned. Despite their importance, these spaces remain poorly understood. We develop a framework for characterizing their topology and geometry using persistent homology, a technique from Topological Data Analysis (TDA) that identifies global structural features such as clusters and voids, and sub-Riemannian geometry, which describes curvature and distance in high-dimensional constrained spaces. Our aim is to bring greater interpretability and theoretical clarity to the internal workings of modern deep learning models.  We apply this framework to two fundamental questions: 1. Edmundson investigates memorization in diffusion models --- generative models that synthesize data by reversing a learned noising process. Memorization, in which a model reproduces training examples rather than generalizing, has significant implications for privacy and robustness. By analyzing how latent space topology and geometry evolve during training, this work seeks structural signatures that are predictive of or diagnostic for memorization. 2. Theisen analyzes the embedding spaces of large language models (LLMs), where words, sentences, and concepts are encoded as geometric vectors. These embedding spaces may be thought of as the LLM analog of the diffusion model's latent space. Using our TDA and geometry tools, this work characterizes how semantic and syntactic structure manifest in these spaces and how organizational patterns vary across architectures and scales. Together, these projects advance our understanding of what deep learning models learn and how that learning is geometrically structured, with implications for interpretability, safety, and model design.