Computable Bounds for Rate Distortion with Feed-Forward for Stationary and Ergodic Sources

In this paper we consider the rate distortion problem of discrete-time, ergodic, and stationary sources with feed forward at the receiver. We derive a sequence of achievable and computable rates that converge to the feed-forward rate distortion. We show that, for ergodic and stationary sources, the rate {align} R_n(D)=\frac{1}{n}\min I(\hat{X}^n\rightarrow X^n){align} is achievable for any $n$, where the minimization is taken over the transition conditioning probability $p(\hat{x}^n|x^n)$ such that $\ex{}{d(X^n,\hat{X}^n)}\leq D$. The limit of $R_n(D)$ exists and is the feed-forward rate distortion. We follow Gallager's proof where there is no feed-forward and, with appropriate modification, obtain our result. We provide an algorithm for calculating $R_n(D)$ using the alternating minimization procedure, and present several numerical examples. We also present a dual form for the optimization of $R_n(D)$, and transform it into a geometric programming problem.