Semidefinite Programming Approach to Gaussian Sequential Rate-Distortion Trade-Offs

Abstract

Abstract-Sequential rate-distortion (SRD) theory provides a framework for studying the fundamental trade-off between datarate and data-quality in real-time communication systems. In this paper, we consider the SRD problem for multi-dimensional timevarying Gauss-Markov processes under mean-square distortion criteria. We first revisit the sensor-estimator separation principle, which asserts that considered SRD problem is equivalent to a joint sensor and estimator design problem in which data-rate of the sensor output is minimized while the estimator's performance satisfies the distortion criteria. We then show that the optimal joint design can be performed by semidefinite programming. A semidefinite representation of the corresponding SRD function is obtained. Implications of the obtained result in the context of zero-delay source coding theory and applications to networked control theory are also discussed.

Semidefinite representation of sequential rate-distortion function for stationary Gauss-Markov processes

“…Notably, the authors of [7] have * Laboratory for Information and Decision Systems, Massachusetts Institute of Technology (Email: ttanaka@mit.edu). This work provides a theoretical support of Section VI-B in [1]. This research was supported by JSPS postdoctoral fellowship.…”

“…Suppose there exists a test channel ⊗ t∈N q(dz t |x t , z t−1 ) ∈ π L satisfying the constraint (2b) such that the corresponding 3 In [1], symbols r and s are used in place of p G and q.…”

“…Without loss of generality, we can assumeÊ t = 0, ∀t ∈ N, since otherwise f q < +∞ is violated (see Claim 4 in [1]). This implies thatẑ t is a linear function of z t−1 .…”

“…This research was supported by JSPS postdoctoral fellowship. 1 In this paper, random variables are denoted by lower-case bold symbols such as x. A multi-dimensional Gaussian random variable x with mean μ and covariance Σ is denoted by x ∼ N(μ, Σ).…”

Zero-Delay Rate Distortion via Filtering for Vector-Valued Gaussian Sources

“…Derpich and Østergaard in [5] considered variants of NRDF for stationary scalar-valued Gaussian autoregressive models of any order with pointwise MSE distortion fidelity and computed the asymptotic expression of the Gaussian NRDF for stationary scalar-valued Gaussian AR(1) sources which was first derived in [17,Equation (1.43)] using alternative methods. Tanaka et al in [19] revisited the finite-time NRDF for vector-valued Gauss-Markov sources with pointwise MSE distortion fidelity following the line of work of [8] and showed that the resulting optimization problem is semidefinite representable, thus, it can be solved numerically. However, none of the previously discussed works, i.e., [5], [8], [18], [19] provide a realization of the test channel conditional distribution that achieves the NRDF or attempt to identify the parameters in the realization given in [17,Theorem 5].…”