Asymptotic Reverse-Waterfilling Characterization of Nonanticipative Rate Distortion Function of Vector-Valued Gauss-Markov Processes with MSE Distortion

Abstract

We analyze the asymptotic nonanticipative rate distortion function (NRDF) of vector-valued Gauss-Markov processes subject to a mean-squared error (MSE) distortion function. We derive a parametric characterization in terms of a reverse-waterfilling algorithm, that requires the solution of a matrix Riccati algebraic equation (RAE). Further, we develop an algorithm reminiscent of the classical reverse-waterfilling algorithm that provides an upper bound to the optimal solution of the reverse-waterfilling optimization problem, and under certain cases, it operates at the NRDF. Moreover, using the characterization of the reverse-waterfilling algorithm, we derive the analytical solution of the NRDF, for a simple twodimensional parallel Gauss-Markov process. The efficacy of our proposed algorithm is demonstrated via an example. P. A. Stavrou and M. Skoglund received funding by the KAW Foundation and the Swedish Foundation for Strategic Research.

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

“…In this paper, we considered certain classes of time-invariant multidimensional Gauss-Markov process which ensure that all matrices in the optimization problem (11) commute by pairs and thus they are simultaneously diagonalizable. As a result of this feature, we showed that the asymptotic reverse-waterfilling characterization of [1] can be simplified considerably and it is equivalent to a reverse-waterfilling algorithm obtained only by the eigenvalues of the involved matrices. For the latter algorithm, we proposed an iterative approach to compute optimally the optimization problem.…”

“…Overall, the previous experiments show that Algorithm 1 is an elegant and more preferable choice to be implemented in modern delay-constrained and computationally-limited devices like the architectures of networked control systems, compared to SDP or [1,Algorithm 1]. It is very fast and adapts extremely well in high dimensional systems (it is scalable).…”

“…(1) The two equivalent forms of the reverse-waterfilling algorithm, i.e., the matrix form and the eigenvalue form, are obtained because in our analysis we prove that for the specific classes of time-invariant Gauss-Markov processes, all matrices commute by pairs hence they are simultaneously diagonalizable by an orthogonal matrix (an information lossless operation). 1 (2) The iterative scheme proposed herein is shown to execute much faster compared to similar numerical approaches that solve the same problem, e.g., [9], [1] and adapts extremely well to high dimensional systems. (3) To the best of the authors knowledge, this is the first non-trivial example of computing asymptotic NRDF beyond scalar-valued Gauss-Markov processes.…”

Bounds on the Sum-Rate of MIMO Causal Source Coding Systems with Memory under Spatio-Temporal Distortion Constraints

“…The most computationally efficient way to compute such problems and, additionally, to gain some insight from the solution is via the well-known reverse-waterfilling algorithm ([ 19 ], Theorem 10.3.3), which is however very hard to construct and compute because one needs to employ and solve complicated KKT conditions [ 37 ]. Such effort was recently made for multivariate Gauss-Markov processes under per instant, averaged total and asymptotically averaged total distortion constraints in [ 24 , 41 ].…”

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

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In this paper, we revisit the asymptotic reversewaterfilling characterization of the nonanticipative rate distortion 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.

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