Lagrangians, Renormalization, and Quantization in Prefix Coding
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912832502628352 |
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| author | Kolpakov, Alexander Rocke, Aidan |
| author_facet | Kolpakov, Alexander Rocke, Aidan |
| contents | We develop a statistical mechanics framework for prefix coding based on variational principles, renormalization, and quantization. A Lagrangian formulation of entropy-optimal encoding under the Kraft-McMillan constraint yields a Gibbs-type implied distribution and completeness of the optimal code. A renormalization operator acting on codeword distribution laws produces a coarse-graining flow whose fixed points have iterated-log structure; discrete quantizations of these fixed points include Elias' $ω$ code as a special case.
Extending the theory to mixed discrete-continuous source laws, we show how continuous codelength functions can be quantized into countable prefix codes and derive resolution-adjusted entropy bounds together with Heisenberg-type and Boltzmann-type relations. This provides a unified and physically motivated view of universal coding, with Elias' $ω$ code as a guiding example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lagrangians, Renormalization, and Quantization in Prefix Coding Kolpakov, Alexander Rocke, Aidan Information Theory Mathematical Physics H.1.1 We develop a statistical mechanics framework for prefix coding based on variational principles, renormalization, and quantization. A Lagrangian formulation of entropy-optimal encoding under the Kraft-McMillan constraint yields a Gibbs-type implied distribution and completeness of the optimal code. A renormalization operator acting on codeword distribution laws produces a coarse-graining flow whose fixed points have iterated-log structure; discrete quantizations of these fixed points include Elias' $ω$ code as a special case. Extending the theory to mixed discrete-continuous source laws, we show how continuous codelength functions can be quantized into countable prefix codes and derive resolution-adjusted entropy bounds together with Heisenberg-type and Boltzmann-type relations. This provides a unified and physically motivated view of universal coding, with Elias' $ω$ code as a guiding example. |
| title | Lagrangians, Renormalization, and Quantization in Prefix Coding |
| topic | Information Theory Mathematical Physics H.1.1 |
| url | https://arxiv.org/abs/2506.23447 |