A Novel Sequential Rate Distortion Function to Compute Partially Observable Markov Processes with MSE Distortion
Photios A. Stavrou, Mikael Skoglund · 2019
We develop a new sequential rate distortion function 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 it is utilized in various applications examples. First, it is used to optimally characterize a finite dimensional optimization problem for partially observable multivariate Gauss-Markov processes and to obtain the optimal linear policies that achieve the solution of this problem. Under the assumption that all matrices commute by pairs, we show that our problem can be cast as a convex optimization problem and achieves its lower rates. We also derive sufficient conditions which ensure that our assumption holds. Then, we compute the optimization problem by solving the KKT conditions and deriving a non-trivial reverse-waterfilling algorithm that we also implement. If our assumption is not met, then, one can still use it to derive sub-optimal waterfilling solutions on the obtained finite dimensional optimization problem. For scalar-valued Gauss-Markov processes with additional observations noise, we derive a new closed form solution and we compared it with the analytical solution obtained for scalar-valued Gauss-Markov processes to infer about the rate loss due to having the additional observations noise. For partially observable time-invariant Markov processes (without observations noise) driven by an additive i.i.d. non-Gaussian system's noise process, we recover using an alternative approach and thus strengthening a recent result by Kostina and Hassibi in [1, Theorem 9] whereas for time-invariant parallel and spatially identically distributed Markov processes driven by additive non-Gaussian noise process we also derive new analytical lower bounds.