<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>

<div>We develop a new sequential rate distortion function</div><div>to compute lower bounds on the average length of all causal prefix free codes for partially observable multivariate Markov processes with mean-squared error distortion constraint. Our information measure is characterized by a variant of causally conditioned directed information and is utilized in various application examples. First, it is used to optimally characterize a finite dimensional optimization problem for jointly Gaussian processes and to obtain the corresponding optimal linear encoding and decoding policies.</div><div>Under the assumption that all matrices commute by pairs,</div><div>we show that our problem can be cast as a convex program</div><div>which achieves its global minimum. We also derive sufficient</div><div>conditions which ensure that our assumption holds. We then</div><div>solve the KKT conditions and derive a new reverse-waterfilling algorithm that we implement. If our assumption is violated, one can still use our approach to derive sub-optimal (upper bound) waterfilling solutions. For scalar-valued Gauss-Markov processes with additional observation noise, we derive a new closed form solution and we compare it with known results in the literature. For partially observable time-invariant Markov processes driven by additive i:i:d: system noise only, we recover using an alternative approach and thus strengthening a recent result by Kostina and Hassibi in [1, Theorem 9] whereas for timeinvariant and spatially IID Markov processes driven by additive noise process we also derive new analytical lower bounds.</div>

<div>We develop a new sequential rate distortion function</div><div>to compute lower bounds on the average length of all causal prefix free codes for partially observable multivariate Markov processes with mean-squared error distortion constraint. Our information measure is characterized by a variant of causally conditioned directed information and is utilized in various application examples. First, it is used to optimally characterize a finite dimensional optimization problem for jointly Gaussian processes and to obtain the corresponding optimal linear encoding and decoding policies.</div><div>Under the assumption that all matrices commute by pairs,</div><div>we show that our problem can be cast as a convex program</div><div>which achieves its global minimum. We also derive sufficient</div><div>conditions which ensure that our assumption holds. We then</div><div>solve the KKT conditions and derive a new reverse-waterfilling algorithm that we implement. If our assumption is violated, one can still use our approach to derive sub-optimal (upper bound) waterfilling solutions. For scalar-valued Gauss-Markov processes with additional observation noise, we derive a new closed form solution and we compare it with known results in the literature. For partially observable time-invariant Markov processes driven by additive i:i:d: system noise only, we recover using an alternative approach and thus strengthening a recent result by Kostina and Hassibi in [1, Theorem 9] whereas for timeinvariant and spatially IID Markov processes driven by additive noise process we also derive new analytical lower bounds.</div>

In this paper we study the problem of characterizing and computing the nonanticipative rate distortion function (NRDF) for partially observable multivariate Gauss-Markov processes with hard mean squared error (MSE) distortion constraints. For the finite time horizon case, we first derive the complete characterization of this problem and its corresponding optimal realization which is shown to be a linear functional of the current time sufficient statistic of the past and current observations signals. We show that when the problem is strictly feasible, it can be computed via semidefinite programming (SDP) algorithm. For time-varying scalar processes with average total MSE distortion we derive an optimal closed form expression by means of a dynamic reverse-waterfilling solution that we also implement via an iterative scheme that convergences linearly in finite time, and a closed-form solution under pointwise MSE distortion constraint. For the infinite time horizon, we give necessary and sufficient conditions to sure that asymptotically the sufficient statistic process of the observation signals achieves a steady-state solution for the corresponding covariance matrices and impose conditions that allow existence of a time-invariant solution. Then, we show that when a finite solution exists in the asymptotic limit, it can be computed via SDP algorithm. We also give strong structural properties on the characterization of the problem in the asymptotic limit that allow for an optimal solution via a reverse-waterfilling algorithm that we implement via an iterative scheme that converges linearly under a finite number of spatial components. Subsequently, we compare the computational time needed to execute for both SDP and reverse-waterfilling algorithms when these solve the same problem to show that the latter is a scalable optimization technique. Our results are corroborated with various simulation studies and are also compared with existing results in the literature.

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. 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.