임선혁 교수님(성균관대학교)

Title : Vietoris-Rips complex, Persistent Homology, and Gromov-Hausdorff distance

김근수 교수님(규슈대학교)

Title: Nonnegative Matrix Factorization with Topological Regularization

Abstract: In this study, we propose Top-NMF, a novel model that incorporates topological regularization into Nonnegative Matrix Factorization (NMF), a widely used dimensionality reduction technique. While conventional regularization methods focus on preserving relationships between data points to guide low-dimensional representations, Top-NMF explicitly controls the topological structure of the support of each basis vector. We interpret each data point as a real-valued function defined over a structured domain (such as a grid or a graph), and treat each basis vector in the same way. Our focus is on the support of each basis vector, and we introduce quantitative topological descriptors derived from persistent homology as regularization terms. These descriptors encourage the support to exhibit desirable properties such as connectedness and modularity. These regularization terms can be applied across diverse domains including time series, images, and graphs and guide the model to learn basis vectors that reflect meaningful structures. We provide a theoretical formulation, describe the optimization scheme, and demonstrate through experiments that Top-NMF achieves structurally faithful and interpretable decompositions.