DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

General single-letter information rate-distortion function

Definition

Assume AC. Let $X, \hat{X}$ be standard-Borel spaces, let $P_{X}$ be a probability law on $X$ , and let $d : X \times \hat{X} \to [0, + \infty]$ be measurable. A feasible joint law, or coupling, is a probability measure $π$ on $X \times \hat{X}$ whose $X$ -marginal is $P_{X}$ . For each distortion budget $D \geq 0$ , define

$R_{s t d} (D) : = \inf_{\begin{array}{c} π_{X} = P_{X} \\ \int d d π \leq D \end{array}} \frac{D (π ∥ P_{X} \otimes π_{\hat{X}})}{\ln 2} = \inf_{\begin{array}{c} π_{X} = P_{X} \\ \int d d π \leq D \end{array}} \frac{I_{n a t} (X; \hat{X})}{\ln 2} .$

The infimum of an empty feasible set is $+ \infty$ . Values are in bits. This definition makes no general existence or attainment claim and supplies no operational coding theorem for arbitrary standard-Borel alphabets.

Depends on

Used by

A Gaussian rate-distortion optimizer need not have a density Counterexample
Finite-dimensional Gaussian vector squared-error rate-distortion function Theorem
Gaussian squared-error rate-distortion function Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Robert M. Gray, Source Coding Theory, first edition (standard reference, not scraped)
Robert M. Gray, Entropy and Information Theory, corrected first edition (standard reference, not scraped)