Information embedding and the triple role of control

TL;DR

This paper explores the problem of embedding information into a Gaussian host signal using control strategies, demonstrating optimality of dirty-paper coding and providing tight bounds related to a classic decentralized control problem.

Contribution

It extends the Gaussian information embedding framework, showing DPC's optimality and deriving tight bounds for the Witsenhausen counterexample in a vector setting.

Findings

DPC achieves optimal rate for perfect host and message recovery.

Bounds show DPC strategies are within a factor of 16 of optimal for MMSE and power.

Improves bounds on the vector Witsenhausen problem to 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 a message, and the decoder recovers both the embedded message and the modified host signal. This partially extends the recent work of Sumszyk and Steinberg to the continuous-alphabet Gaussian setting. Through a control-theoretic lens, we observe that the problem is a minimalist example of what is called the "triple role" of control actions. We show that a dirty-paper-coding strategy achieves the optimal rate for perfect recovery of the modified host and the message for any message rate. For imperfect recovery of the modified host, by deriving bounds on the minimum mean-square error (MMSE) in recovering the modified host signal, we show that DPC-based strategies are guaranteed to attain within a uniform constant factor of 16 of the optimal weighted sum of power required in host signal modification and the MMSE in the modified host signal reconstruction for all weights and all message rates. 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 decentralized control: the Witsenhausen counterexample. Numerically, this tighter bound helps us characterize the asymptotically optimal costs for the vector Witsenhausen problem to within a factor of 1.3 for all problem parameters, improving on the earlier best known bound of 2.