← 研踪 PaperTrailarXiv:2611.16253 · v1
A Constructive Theory of Closed-Form Rate-Distortion Functions
首次提交 2026-11-12 11:56
版本更新 2026-11-12 11:56
雷达观测 2026-11-13 14:16
摘要
Rate-distortion functions are defined by a variational problem whose analytic solution depends on the source distribution and distortion measure. We develop a posterior-based construction framework using conditional entropy as a common distortion coordinate. Building on the rate-entropy formulation, we establish entropy-level attainability for discrete sources with finite entropy and absolutely continuous sources with integrable log densities. We give a constructive rate-entropy coding proof for finite-alphabet sources and such continuous sources, without a second-moment assumption. Marginal compatibility ensures that a candidate posterior and output distribution recover the source, while a Gibbs representation and a kernel mass bound certify attainment of maximum conditional entropy. The negative log posterior identifies candidate fixed distortions. Excess rate equals a conditional entropy gap and decomposes into budget slack and posterior mismatch. Convolution decompositions and conjugate posteriors provide two construction paths. They recover established results, including the Gamma-source rate-distortion function under Itakura-Saito distortion, and yield a closed-form parametric expression for zeta sources under divisibility-constrained logarithmic multiplicative distortion. Exact expressions are obtained on specified intervals for negative binomial sources under one-sided counting distortion and Gamma sources under one-sided mean error. A residual-count coordinate for Poisson entropy budgets and rate-entropy parametrizations of Gamma and Dirichlet counting channels connect model parameters to remaining uncertainty. The framework unifies distortion selection, compatible-channel construction, and optimality verification.
作者备注：26 pages, 3 figures, 1 table
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v1 · 2026-11-12 11:56 · 最新观测版本
A Constructive Theory of Closed-Form Rate-Distortion Functions
Rate-distortion functions are defined by a variational problem whose analytic solution depends on the source distribution and distortion measure. We develop a posterior-based construction framework using conditional entropy as a common distortion coordinate. Building on the rate-entropy formulation, we establish entropy-level attainability for discrete sources with finite entropy and absolutely continuous sources with integrable log densities. We give a constructive rate-entropy coding proof for finite-alphabet sources and such continuous sources, without a second-moment assumption. Marginal compatibility ensures that a candidate posterior and output distribution recover the source, while a Gibbs representation and a kernel mass bound certify attainment of maximum conditional entropy. The negative log posterior identifies candidate fixed distortions. Excess rate equals a conditional entropy gap and decomposes into budget slack and posterior mismatch. Convolution decompositions and conjugate posteriors provide two construction paths. They recover established results, including the Gamma-source rate-distortion function under Itakura-Saito distortion, and yield a closed-form parametric expression for zeta sources under divisibility-constrained logarithmic multiplicative distortion. Exact expressions are obtained on specified intervals for negative binomial sources under one-sided counting distortion and Gamma sources under one-sided mean error. A residual-count coordinate for Poisson entropy budgets and rate-entropy parametrizations of Gamma and Dirichlet counting channels connect model parameters to remaining uncertainty. The framework unifies distortion selection, compatible-channel construction, and optimality verification.
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