# Vector Gaussian CEO Problem Under Logarithmic Loss

## TL;DR

This paper characterizes the exact rate-distortion region for the vector Gaussian CEO problem under logarithmic loss, using Fisher information and de Bruijn identity, and shows Gaussian test channels are optimal.

## Contribution

It provides the first explicit characterization of the rate-distortion region for this problem under logarithmic loss, extending previous quadratic distortion results.

## Key findings

- Explicit rate-distortion region for the vector Gaussian CEO problem under logarithmic loss.
- Gaussian test channels with time-sharing are optimal for the Berger-Tung inner bound.
- The result offers an outer bound for a quadratic vector Gaussian CEO problem with determinant constraint.

## Abstract

In this paper, we study the vector Gaussian Chief Executive Officer (CEO) problem under logarithmic loss distortion measure. Specifically, $K \geq 2$ agents observe independently corrupted Gaussian noisy versions of a remote vector Gaussian source, and communicate independently with a decoder or CEO over rate-constrained noise-free links. The CEO wants to reconstruct the remote source to within some prescribed distortion level where the incurred distortion is measured under the logarithmic loss penalty criterion. We find an explicit characterization of the rate-distortion region of this model. For the proof of this result, we obtain an outer bound on the region of the vector Gaussian CEO problem by means of a technique that relies on the de Bruijn identity and the properties of Fisher information. The approach is similar to Ekrem-Ulukus outer bounding technique for the vector Gaussian CEO problem under quadratic distortion measure, for which it was there found generally non-tight; but it is shown here to yield a complete characterization of the region for the case of logarithmic loss measure. Also, we show that Gaussian test channels with time-sharing exhaust the Berger-Tung inner bound, which is optimal. Furthermore, we also show that the established result under logarithmic loss provides an outer bound for a quadratic vector Gaussian CEO problem with determinant constraint, for which we characterize the optimal rate-distortion region.