Information embedding meets distributed control

TL;DR

This paper extends information embedding to Gaussian signals, demonstrating optimal strategies for message and host recovery, and provides new bounds and approximations for the vector Witsenhausen problem in distributed control.

Contribution

It introduces a dirty-paper coding approach for Gaussian information embedding and derives the tightest bounds for the vector Witsenhausen problem to date.

Findings

Optimal rate achieved by dirty-paper coding for perfect recovery

Bounds established for partial host recovery within specified distortion

Numerical characterization of Witsenhausen costs within a factor of 1.3

Abstract

We consider the problem of information embedding where the encoder modifies a white Gaussian host signal in a power-constrained manner to encode the message, and the decoder recovers both the embedded message and the modified host signal. This extends the recent work of Sumszyk and Steinberg to the continuous-alphabet Gaussian setting. We show that a dirty-paper-coding based strategy achieves the optimal rate for perfect recovery of the modified host and the message. We also provide bounds for the extension wherein the modified host signal is recovered only to within a specified distortion. When specialized to the zero-rate case, our results provide the tightest known lower bounds on the asymptotic costs for the vector version of a famous open problem in distributed control -- the Witsenhausen counterexample. Using this bound, we characterize the asymptotically optimal costs for the vector Witsenhausen problem numerically to within a factor of 1.3 for all problem parameters, improving on the earlier best known bound of 2.