  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.
