Rate distortion theory treats the problem of encoding a source with minimum codebook size while at the same time allowing for a certain amount of errors in the reconstruction measured by a fidelity criterion and distortion level. Similar to the channel coding problem the optimal rate of the codebook with respect to the blocklength is given by a convex optimization problem involving information theoretic quantities like mutual information. The value of the rate in dependence of the distortion level as well as the optimizer used in the codebook construction are of theoretical and practical importance in communication and information theory. In this paper the behavior of the rate distortion function regarding the computability of the optimizing test channel is investigated. We find that comparable with known results about the optimizer for other information theoretic problems a similar result is found to be true also regarding the computability of the optimizer for rate distortion functions. It turns out that while the rate distortion function is usually computable the optimizer for this problem is in general non-computable even for simple distortion measures.

翻译：率失真理论研究在允许重建过程中存在一定误差（通过保真度准则和失真水平衡量）的前提下，以最小码本规模对信源进行编码的问题。与信道编码问题类似，码本相对于块长度的最优速率可通过涉及互信息等信息论量的凸优化问题给出。失真水平依赖的速率值以及码本构造中使用的优化器在通信与信息理论中具有重要的理论和实践意义。本文研究了率失真函数关于优化测试信道可计算性的行为。我们发现，与已知的其他信息论问题优化器的结果相比，关于率失真函数优化器的可计算性也存在类似结论。结果表明，虽然率失真函数通常是可计算的，但即使对于简单的失真度量，该问题的优化器在一般情况下也是不可计算的。