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.

Finite-alphabet information rate-distortion function

Definition

Let $X, \hat{X}$ be nonempty finite alphabets, let $p$ be a pmf on $X$ , and let $d : X \times \hat{X} \to [0, \infty)$ be finite-valued. A test channel is a stochastic matrix $W (\hat{x} ∣ x)$ , and its expected single-letter distortion is $\sum_{x, \hat{x}} p (x) W (\hat{x} ∣ x) d (x, \hat{x})$ , with block distortion as in Single-letter and block distortion. Put $D_{\min} = \sum_{x} p (x) \min_{\hat{x}} d (x, \hat{x})$ . Define, in bits,

R_{p, d} (D) : = {\begin{cases} \min_{W : E d (X, \hat{X}) \leq D} \frac{I (X; \hat{X})}{\ln 2}, & D \geq D_{\min}, \\ + \infty, & D < D_{\min} . \end{cases}

The minimum exists by Attainment of finite-alphabet rate-distortion minima. Values are not operational code rates; the finite memoryless coding theorem is a separate claim.

Depends on

Used by

Dependency tree · two levels

14 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

MIT OpenCourseWare 6.441, Chapter 23: Rate-distortion theory (standard reference, not scraped)
Robert M. Gray, Source Coding Theory, first edition (standard reference, not scraped)