A Universal Random Coding Ensemble for Sample-Wise Lossy Compression

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We propose a universal ensemble for the random selection of rate–distortion codes which is asymptotically optimal in a sample-wise sense. According to this ensemble, each reproduction vector, (Formula presented.), is selected independently at random under the probability distribution that is proportional to (Formula presented.), where (Formula presented.) is the code length of (Formula presented.) pertaining to the 1978 version of the Lempel–Ziv (LZ) algorithm. We show that, with high probability, the resulting codebook gives rise to an asymptotically optimal variable-rate lossy compression scheme under an arbitrary distortion measure, in the sense that a matching converse theorem also holds. According to the converse theorem, even if the decoder knew the ℓ-th order type of source vector in advance (ℓ being a large but fixed positive integer), the performance of the above-mentioned code could not have been improved essentially for the vast majority of codewords pertaining to source vectors in the same type. Finally, we present a discussion of our results, which includes among other things, a clear indication that our coding scheme outperforms the one that selects the reproduction vector with the shortest LZ code length among all vectors that are within the allowed distortion from the source vector.

Original languageEnglish
Article number1199
Issue number8
StatePublished - Aug 2023


  • LZ algorithm
  • code ensemble
  • cumulant generating function
  • lossy compression
  • random coding
  • rate–distortion
  • source coding
  • universal coding
  • universal distribution

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Electrical and Electronic Engineering
  • General Physics and Astronomy
  • Mathematical Physics
  • Physics and Astronomy (miscellaneous)


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