Caching (a pair of) Gaussians

A source produces i.i.d. vector samples from a Gaussian distribution, but the user is interested in only one component. In the cache phase, not knowing which component the user is interested in, a first compressed description is produced. Upon learning the user’s choice, a second message is provided in the update phase so as to attain the desired fidelity on that component. We aim to find the cache strategy that minimizes the average update rate. We show that for Gaussian codebooks, the optimal strategy depends on whether or not the cache is large enough to make the vector conditionally independent. If it is, infinitely many equally optimal strategies exist. If it is not, we show that the encoder should project the source onto some subspace prior to coding. For a pair of Gaussians, we exactly characterize this projection vector.