Rate Region of the Gaussian Scalar-Help-Vector Source-Coding Problem

We determine the rate region of the Gaussian one-helper source-coding problem in which the helper observes a scalar, the main encoder observes a vector, and the distortion constraint is a positive semidefinite upper bound on the error covariance matrix of the main source. The rate region is achieved by a Gaussian achievable scheme. We introduce a novel outer bounding technique to establish the converse. Our approach is to create a reduced dimensional problem by projecting the main source and the distortion constraint in certain directions determined by the optimal Gaussian scheme. We also provide an outer bound to the rate region of the more general problem in which there are distortion constraints on both sources. This outer bound is partially tight in general and completely tight in some nontrivial cases.