Computing the optimal error exponential function for fixed-length lossy coding in discrete memoryless sources

Abstract: 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.