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 source distribution $P$ and the distortion measure $d(x,y)$. The reason for the discontinuity in the error exponent is that there exists a distortion measure $d(x,y)$ and a distortion level $\Delta$ 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. A major contribution of this paper is to provide a parametric representation that perfectly matches the inverse function of Marton's exponent, thereby preventing the problems arising from the above-mentioned non-concavity of $R(\Delta|P)$. For fixed parameters, an optimal distribution can be obtained using Arimoto's algorithm. By performing a nonconvex optimization over the parameters, the inverse function of Marton's exponent is obtained.

翻译：固定长度有损信源编码的误差指数由Marton建立。Ahlswede证明，该误差指数可能在速率$R$处出现不连续性，具体取决于信源分布$P$和失真测度$d(x,y)$。误差指数不连续性的原因在于存在失真测度$d(x,y)$和失真水平$\Delta$，使得率失真函数$R(\Delta|P)$相对于$P$既非凹函数也非拟凹函数。用于计算有损信源编码误差指数的Arimoto算法基于Blahut对误差指数的参数化表示。然而，Blahut的参数化表示是Marton指数的一个下凸包络，两者通常并不一致。本文的主要贡献在于提供了一种参数化表示，该表示完美匹配Marton指数的逆函数，从而避免了因上述$R(\Delta|P)$非凹性引发的问题。对于固定参数，可通过Arimoto算法获得最优分布。通过对参数进行非凸优化，即可得到Marton指数的逆函数。