Computation of the optimal error exponent function for fixed-length lossy source coding in discrete memoryless sources

Marton's optimal error exponent for the lossy source coding problem is defined as a non-convex optimization problem. This fact had prevented us to develop an efficient algorithm to compute it. This problem is caused by the fact that the rate-distortion function $R(\Delta|P)$ is potentially non-concave in the probability distribution $P$ for a fixed distortion level $\Delta$. The main contribution of this paper is the development of a parametric expression that is in perfect agreement with the inverse function of the Marton exponent. This representation has two layers. The inner layer is convex optimization and can be computed efficiently. The outer layer, on the other hand, is a non-convex optimization with respect to two parameters. We give a method for computing the Marton exponent based on this representation.