The Time-Invariant Multidimensional Gaussian Sequential Rate-Distortion Problem Revisited

We revisit the sequential rate-distortion (SRD) tradeoff problem for vector-valued Gauss-Markov sources with mean-squared error distortion constraints. Our study is partly motivated by the question recently raised in the paper “Rate-cost tradeoffs in control” (in Proc. 54th Annu. Allerton Conf. Commun., Control, Comput., 2016, pp. 1157-1164) regarding the correctness of its solution algorithm known in the literature. We show via a counterexample that the dynamic reverse water-filling algorithm suggested by (15) of the paper “Stochastic linear control over a communication channel” (IEEE Trans. Autom. Control, vol. 49, pp. 1549-1561, 2004) is not applicable to this problem, and consequently, the closed-form expression of the asymptotic SRD function derived in (17) of the paper “Stochastic linear control over a communication channel” (IEEE Trans. Autom. Control, vol. 49, pp. 1549-1561, 2004) is not correct in general. Nevertheless, we show that the multidimensional Gaussian SRD function is semidefinite representable, and thus, it is readily computable.