Optimal zero-delay coding (quantization) of $\mathbb{R}^d$-valued linearly generated Markov sources is studied under quadratic distortion. The structure and existence of deterministic and stationary coding policies that are optimal for the infinite horizon average cost (distortion) problem are established. Prior results studying the optimality of zero-delay codes for Markov sources for infinite horizons either considered finite alphabet sources or, for the $\mathbb{R}^d$-valued case, only showed the existence of deterministic and non-stationary Markov coding policies or those which are randomized. In addition to existence results, for finite blocklength (horizon) $T$ the performance of an optimal coding policy is shown to approach the infinite time horizon optimum at a rate $O(\frac{1}{T})$. This gives an explicit rate of convergence that quantifies the near-optimality of finite window (finite-memory) codes among all optimal zero-delay codes.
翻译:在二次扭曲的情况下,研究了对无限地平线平均成本(扭曲)问题最理想的确定性和固定性编码政策的结构和存在。先前研究无限地平线源的Markov源零延迟编码最佳性(量化)的结果,或者被认为是有限的字母源,或者,对于$mathbb{R ⁇ d$估值的案例中,仅显示存在确定性和非静止的Markov编码政策或随机化的政策。除了存在结果外,对于有限区块长度(horison)而言,最佳编码政策的性能显示为以$O(\frac{1 ⁇ T}$接近无限时间范围的最佳性。这提供了一种明确的趋同率,使所有最佳零缓冲代码中有限的窗口(fite-emory)的接近最佳性(fite-emory)代码四分化为四分化。