The error exponent of fixed-length lossy source coding was established by Marton. Ahlswede showed that this exponent can be discontinuous at a rate $R$, depending on the probability distribution $P$ of the given information source and the distortion measure $d(x,y)$. The reason for the discontinuity in the error exponent is that there exists $(d,\Delta)$ such that the rate-distortion function $R(\Delta|P)$ is neither concave nor quasi-concave with respect to $P$. Arimoto's algorithm for computing the error exponent in lossy source coding is based on Blahut's parametric representation of the error exponent. However, Blahut's parametric representation is a lower convex envelope of Marton's exponent, and the two do not generally agree. The contribution of this paper is to provide a parametric representation that perfectly matches with the inverse function of Marton's exponent, thus avoiding the problem of the rate-distortion function being non-convex with respect to $P$. The optimal distribution for fixed parameters can be obtained using Arimoto's algorithm. Performing a nonconvex optimization over the parameters successfully yields the inverse function of Marton's exponent.

翻译：定长有损信源编码的误差指数由Marton建立。Ahlswede指出，该误差指数在码率$R$处可能不连续，具体取决于给定信息源的概率分布$P$和失真度量$d(x,y)$。误差指数不连续的原因在于存在$(d,\Delta)$使得率失真函数$R(\Delta|P)$关于$P$既非凹函数也非拟凹函数。Arimoto算法计算有损信源编码中误差指数的方法基于Blahut对该误差指数的参数化表示。然而，Blahut的参数化表示是Marton指数的下凸包络，两者通常不一致。本文的贡献在于提出了一种与Marton指数反函数完全匹配的参数化表示，从而避免了率失真函数关于$P$非凸的问题。固定参数下的最优分布可通过Arimoto算法获得。对参数进行非凸优化后，即可成功得到Marton指数的反函数。