Asymptotic Reverse Waterfilling Algorithm of NRDF for Certain Classes of Vector Gauss-Markov Processes

Abstract

<div>In this paper, we revisit the asymptotic reverse-waterfilling characterization of the nonanticipative rate distortion</div><div>function (NRDF) derived for a time-invariant multidimensional Gauss-Markov processes with mean-squared error (MSE) distortion in [1]. We show that for certain classes of time-invariant multidimensional Gauss-Markov processes, the specific characterization behaves as a reverse-waterfilling algorithm obtained in matrix form ensuring that the numerical approach of [1, Algorithm 1] is optimal. In addition, we give an equivalent characterization that utilizes the eigenvalues of the involved matrices reminiscent of the well-known reverse-waterfilling algorithm in information theory. For the latter, we also propose a novel numerical approach to solve the algorithm optimally. The efficacy of our proposed iterative scheme compared to similar existing schemes is demonstrated via experiments. Finally, we use our new results to derive an analytical solution of the asymptotic NRDF for a correlated time-invariant two-dimensional Gauss-Markov process.</div>

Indirect NRDF for Partially Observable Gauss-Markov Processes with MSE Distortion: Complete Characterizations and Optimal Solutions

“…(2) If in (44), we take µ Q,i = 0, for some i, then for that i, it can be easily shown that our solution will recover as a special case, the reverse-waterfilling algorithm obtained for the "fully observable" time-invariant multidimensional Gauss-Markov processes subject to a MSE distortion constraint derived in [24,Proposition 1].…”

“…(3) Due to the pay-off in (44) that is greater than the corresponding one obtained for fully observable processes (only the first term appears in that case), the reverse-waterfilling solution of Theorem 5 will always be an upper bound on the reverse-waterfilling obtained for fully observable multivariate processes in [24,Proposition 1] for the rate-distortion region that these are both defined.…”

“…In such case, the tightness of our bound depends on the structure of the given matrices and the dimensionality of the partially observable system. We note that the sub-optimality of our algorithm for arbitrarily chosen matrices comes with the advantage of an algorithmic approach that is extremely fast and easily adaptable to high dimensional systems when implemented (see, e.g., [24] for more details). Our result in (R5) is compared with an existing lower bound obtained in [25, Appendix A] using [1] to show via simulation experiments that the latter is not tight, in general, with respect to its optimal solution.…”

Indirect NRDF for Partially Observable Gauss-Markov Processes with MSE Distortion: Complete Characterizations and Optimal Solutions

“…where θ * > 0, µ Υ,i 2µ A 2 ,i µΣ ,i > 0 5 and D > D min [0,∞] . Proof: When Σ 0 the derivation follows using similar steps of the derivation of [17,Theorem 2] and we omit it. However, if for instance in Proposition 3, (i), Σ 0 we use a standard continuity argument, that is, there exists a δ > 0 such that Σ = Σ + I p is nonsingular for all ∈ (0, δ) (see, e.g., [36,Theorem 2.9]).…”

“…Another important question has to do with the conditions that are needed to ensure (strict) feasibility of the optimization problem in both finite and infinite time horizon. Equally important questions include the derivation of optimal or suboptimal (numerical or analytical) solutions for this problem for both scalar or beyond scalar processes as well as the analysis of the problem for high dimensional systems that necessitates scalable optimization algorithms (an issue already known from the analysis of [17]).…”