An Upper Bound of Cumulant Generating Function of Codeword Lengths in Variable-Length Lossy Source Coding Under Logarithmic Loss

This paper deals with variable-length lossy source coding. We consider a single-shot approach to source coding in which the source to be compressed is a random variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex>. The performance criterion of codeword lengths is a cumulant generating function of codeword lengths and the performance criterion of a distortion measure is a logarithmic loss distortion. We derive an upper bound of the cumulant generating function of codeword lengths under the condition that the logarithmic loss distortion is less than or equal to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$D$</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$D\geq 0$</tex> is a given distortion level. The upper bound is characterized by the Rényi entropy of the random variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$[\frac{X}{\lfloor\exp(D)\rfloor}]$</tex>. Numerical examples show that there are cases where the new upper bound is tighter than the upper bound derived in a previous study.