The CEO Problem With Inter-Block Memory

An <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-dimensional source with memory is observed by <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> isolated encoders via parallel channels, who compress their observations to transmit to the decoder via noiseless rate-constrained links while leveraging their memory of the past. At each time instant, the decoder receives <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> new codewords from the observers, combines them with the past received codewords, and produces a minimum-distortion estimate of the latest block of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> source symbols. This scenario extends the classical one-shot CEO problem to multiple rounds of communication with communicators maintaining the memory of the past. We extend the Berger-Tung inner and outer bounds to the scenario with inter-block memory, showing that the minimum asymptotically (as <inline-formula> <tex-math notation="LaTeX">$n \to \infty $ </tex-math></inline-formula>) achievable sum rate required to achieve a target distortion is bounded by minimal directed mutual information problems. For the Gauss-Markov source observed via <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> parallel AWGN channels, we show that the inner bound is tight and solve the corresponding minimal directed mutual information problem, thereby establishing the minimum asymptotically achievable sum rate. Finally, we explicitly bound the rate loss due to a lack of communication among the observers; that bound is attained with equality in the case of identical observation channels. The general coding theorem is proved via a new nonasymptotic bound that uses stochastic likelihood coders and whose asymptotic analysis yields an extension of the Berger-Tung inner bound to the causal setting. The analysis of the Gaussian case is facilitated by reversing the channels of the observers.