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.

Couplings of finite probability vectors

Definition

Let $X = {x_{1}, \dots, x_{m}}$ be a finite nonempty set with this fixed ordering, and let $p = (p_{i})_{i = 1}^{m}$ and $q = (q_{j})_{j = 1}^{m}$ be probability vectors on $X$ : their coordinates are nonnegative real numbers and $\sum_{i = 1}^{m} p_{i} = \sum_{j = 1}^{m} q_{j} = 1$ . A coupling of $p$ and $q$ is a real matrix $π = (π_{i j})_{i, j = 1}^{m}$ satisfying

π_{i j} \geq 0, \sum_{j = 1}^{m} π_{i j} = p_{i} (1 \leq i \leq m), \sum_{i = 1}^{m} π_{i j} = q_{j} (1 \leq j \leq m) .

Write $Γ (p, q)$ for the set of all such matrices. The first index records the source point and the second index records the destination point.

Definition references

Finite probability vectors

Peyré and Cuturi, Computational Optimal Transport, §§2.1–2.3 and 3.1

Santambrogio, Introduction to Optimal Transport Theory, §1.1

Depends on

Finite probability vectors

Used by

Dependency tree · one level

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Sources

Gabriel Peyré and Marco Cuturi, Computational Optimal Transport, §§2.1–2.3 and 3.1 (standard reference, not scraped)
Filippo Santambrogio, Introduction to Optimal Transport Theory, §1.1, PDF pp. 3–4 (standard reference, not scraped)