The Optimal Compression Rate of Variable-to-Fixed Length Source Coding with a Non-Vanishing Excess-Distortion Probability

We consider the variable-to-fixed length lossy source coding (VFSC) problem. The optimal compression rate of the average length of variable-to-fixed source coding, allowing a non-vanishing probability of excess-distortion $\varepsilon$, is shown to be equal to $(1-\varepsilon)R(D)$, where $R(D)$ is the rate-distortion function of the source. In comparison to the related results of Koga and Yamamoto as well as Kostina, Polyanskiy, and Verd\'{u} for fixed-to-variable length source coding, our results demonstrate an interesting feature that variable-to-fixed length source coding has the same first-order compression rate as fixed-to-variable length source coding.