AlgorithmAlgorithm: AI-adaptedProof: 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.

Exhaustive global empirical squared-error quantizer

Definition

Input: integers $N, d, k \geq 1$ , indexed data $x_{1}, \dots, x_{N} \in R^{d}$ , and exact positive weights $w_{i}$ with $\sum_{i} w_{i} = 1$ . Enumerate all assignments $a = (a_{1}, \dots, a_{N}) \in {1, \dots, k}^{N}$ in lexicographic order. For each assignment, let $S_{j} (a) = {i : a_{i} = j}$ . For a nonempty $S_{j} (a)$ set

$c_{j} (a) : = \frac{\sum_{i \in S_{j} (a)} w_{i} x_{i}}{\sum_{i \in S_{j} (a)} w_{i}};$

for an empty $S_{j} (a)$ set $c_{j} (a) : = x_{1}$ . Compute the exact assigned cost $L (a) : = \sum_{i} w_{i} ∥ x_{i} - c_{a_{i}} (a) ∥_{2}^{2}$ . Retain the first assignment with minimal cost and its center tuple. Return that tuple together with the least-index nearest-center assignment for it. The finite enumeration, tie rule and empty-cluster convention are part of the algorithm.

Depends on

Used by

Exhaustive empirical search attains a global quantizer Theorem

Dependency tree · two levels

4 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 Gallager, 6.450 Principles of Digital Communications I, Chapter 3: Quantization (standard reference, not scraped)
Robert M. Gray and David L. Neuhoff, Quantization, IEEE Transactions on Information Theory 44(6) (1998) (standard reference, not scraped)