On the Geometry and Quantization of Manifolds of Positive Semi-Definite Matrices

The geometry of different spaces of positive semi-definite matrices buffeted by rank and trace constraints is studied. In addition to revealing their Riemannian structure, we derive the normalized volume of a ball over these spaces. Further, we use the leading coefficient from the ball volume expansion to bound the quantization error incurred with finite-sized sphere-packing codebooks as well as random codebooks to represent sources distributed over general Riemannian manifolds.