  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.
