The CEO problem

The CEO problem is a multiterminal source coding problem involving estimation or detection in networks and was introduced in. The set up of the problem involves a Chief Executive Officer (CEO) who employs multiple agents observing noisy observations of an underlying source. The agents communicate their observations through rate constrained links to the CEO. Alternately, one could also interpret this situation as though the CEO is limited by the amount of the data that he or she can interpret and therefore the agents need to send limited information. As a result, the agents need to compress their observations and send them to the CEO. Given that the CEO is willing to allow for a certain distortion in the estimate of the underlying source, the objective of the problem is to find out what are the smallest rates at which the CEO will be able to estimate the source satisfying a desired distortion criterion. While an exact characterization of this problem is not known, the Gaussian version of this problem known as the Quadratic Gaussian CEO problem has been solved.

Quadratic Gaussian CEO Problem
The quadatic Gaussian CEO problem represents a version of the CEO problem in which a Gaussian source corrupted by Gaussian noise is observed by the encoders. Given a squared error criterion on the allowable distortion at the decoder, the sum rate and the rate region of this problem were solved in and  respectively. The optimal achievability scheme is given by quantization followed by random binning at each encoder while the outer bound involves parameterizing the code and formulating a convex optimization problem in terms of these parameters. The key aspect of this parametrization is the rate of quantization of the noise at each encoder. The outer bound optimization dictates this rate of quantization of the noise which in turn leads to the optimal overall quantization rate at each encoder. The quantize and bin strategy is a simple way of expressing the Berger-Tung achievable scheme for multiterminal source coding. In the case of the Gaussian CEO problem, the optimal rate region is achieved by choosing Gaussian auxiliaries in the Berger-Tung achievable scheme.

Let $\{X_i\}_{i=1}^n$ be an i.i.d. realizations of a Gaussian source with mean zero and variance $\sigma_X^2$. $L$ encoders or agents observe noisy versions of this source given by $\{Y_{li}\}_{i=1}^n$ for $l\in\{1,2,\ldots,L\}$, where $Y_{li}=X_{li}+N_{li}$, $\{N_{li}\}_{i=1}^n$ with $N_{li}\sim\mathcal{N}(0,\sigma_{N_l}^2)$ and are independent across time and the agents. Each encoder applies an encoding function $f_l$ on the observed sequence resulting in a codeword $C_l\in\mathcal{C}_l$. The rate of the encoding process at each encoder is given by $R_l=\frac{1}{n}\log |\mathcal{C}_l|$. The decoder applies a decoding function $\phi$ on the received codewords $C_1,C_2,\dots,C_L$ resulting in an estimate of the source sequence, $\{\hat{X}_i\}_{i=1}^n$. The distortion suffered in this process is given by \begin{equation*} D = \sum_{i=1}^n\mathbb{E}\left[(X_{i}-\hat{X}_{i})^2\right]. \end{equation*}

The rate distortion region for this problem is given by the set of all tuples $(R_1,R_2,\dots,R_L,D)$ such that there exist encoding functions $f_1,f_2,\ldots,f_L$ and decoding function $\phi$ such that Encoder $l$ achieves a rate $R_l$ and the decoder estimates the source to within a distortion $D$.

Main Result
Let \begin{equation*} \mathcal{F}(D) = \left\{(r_1,r_2,\ldots,r_L) \in\mathbb{R}_L^{+} : \frac{1}{\sigma_X^2} + \sum_{l=1}^L\frac{1-e^{-2r_l}}{\sigma_{N_l}^2} = \frac{1}{D}\right\}. \end{equation*}

The rate distortion region for the CEO problem is given by the set

\begin{equation*} \mathcal{R}(r_1,r_2,\ldots,r_L) = \bigcup_{(r_1,r_2,\ldots,r_L)\in\mathcal{F}(D)}\left\{\sum_{l\in A}R_l \geq \frac{1}{2}\log\frac{\sigma_X^2}{D}+\sum_{l\in A}r_l - \sum_{l\in A^c}\frac{1-e^{-2r_l}}{\sigma_{N_l}^2} \forall A\neq \Phi \textrm{ and } A\subseteq\{1,2,\ldots,L\}\right\}. \end{equation*}