Data analysis using Riemannian geometry and applications to chemical engineering. (December 2022) - Smith, Alexander; Laubach, Benjamin; Castillo, Ivan; Zavala, Victor M.
Abstract: We explore the use of tools from Riemannian geometry for the analysis of symmetric positive definite matrices (SPD). An SPD matrix is a versatile data representation that is commonly used in chemical engineering (e.g., covariance/correlation/Hessian matrices and images) and powerful techniques are available for its analysis (e.g., principal component analysis). A key observation that motivates this work is that SPD matrices live on a Riemannian manifold and that implementing techniques that exploit this basic property can yield significant benefits in data-centric tasks such as classification and dimensionality reduction. We demonstrate this via a couple of case studies that conduct anomaly detection in the context of process monitoring and image analysis. Highlights: Mathematical introduction to Riemannian manifolds and their geometric analysis. Framework (and code) for the analysis of SPD matrices through Riemannian geometry. Benefits of geometric approach to data analysis illustrated through two real world case studies.