Quantization on the Grassmann Manifold

The problem of quantization in an Euclidean space with unitary constraints can be formulated as an unconstrained problem on a Grassmann manifold. Such constraints arise in areas such as wireless communication with multiple antennas at the transmitter and at the receiver. Due to the constraints, the distortion rate analysis developed for Euclidean spaces cannot be applied directly. This paper extends Gersho's asymptotic (large rate, small distortion) distortion bounds to the case when the source is distributed on the complex Grassmann manifold. The special structure of the Grassmann manifold and the distortion measures defined on it differentiate this problem from the traditional vector quantization in Euclidean spaces.