RIEMANNIAN OPTIMIZATION OVER PSD FIXED-RANK MATRIX CONSTRAINTS AND APPLICATIONS

<p dir="ltr">This dissertation develops a comprehensive geometric framework for optimization and sampling over the manifold of fixed-rank Hermitian positive semidefinite (PSD) matrices, motivated by applications in signal processing, machine learning, and inverse problems. Using tools from Riemannian geometry, we formulate both embedded and quotient manifold representations and analyze their curvature, tangent spaces, and metric structures. The thesis introduces several Riemannian optimization algorithms—including conjugate gradient and orthogonalization-free methods—designed specifically for fixed-rank PSD constraints.</p><p dir="ltr">A key focus is the effect of rank-deficiency on the conditioning of Riemannian Hessians under various metrics. To address this, we perform a detailed condition number analysis and validate our findings through numerical experiments on eigenvalue problems, matrix completion, phase retrieval, and interferometric inversion.</p><p dir="ltr">In addition to optimization, we extend our framework to sampling. We construct Riemannian Langevin Monte Carlo schemes for fixed-rank PSD matrices under both embedded and quotient geometries. These methods allow for Gibbs sampling on manifolds and are validated using test functions with known distributions.</p><p dir="ltr">All experiments are reproducible using the algorithms and geometric operators defined in this work. No human subjects or personal data were involved, so no ethical approval was required. The techniques presented are general and can be extended to other manifold-constrained problems in computational mathematics and engineering.</p>