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.

The finite-metric 1-Wasserstein transport cost

Definition

Let $X = {x_{1}, \dots, x_{m}}$ be a nonempty finite set and let $d : X \times X \to R$ satisfy $d (x, y) = 0$ if and only if $x = y$ , $d (x, y) = d (y, x)$ , and $d (x, z) \leq d (x, y) + d (y, z)$ . Let $p = (p_{i})_{i = 1}^{m}$ and $q = (q_{i})_{i = 1}^{m}$ be probability vectors, so their coordinates are nonnegative and each vector sums to $1$ . A coupling is an $m \times m$ matrix $π$ with $π_{i j} \geq 0$ , row sums $\sum_{j} π_{i j} = p_{i}$ , and column sums $\sum_{i} π_{i j} = q_{j}$ ; write $Γ (p, q)$ for the set of all such matrices. Writing $d_{i j} = d (x_{i}, x_{j})$ , define the finite-metric 1-Wasserstein transport cost by

W_{1} (p, q) : = \min_{π \in Γ (p, q)} \sum_{i = 1}^{m} \sum_{j = 1}^{m} d_{i j} π_{i j} .

Definition references

Metric spaces

Finite probability vectors

Couplings of finite probability vectors

Peyré and Cuturi, Computational Optimal Transport, §§2.1–2.3 and 6.1

Santambrogio, Introduction to Optimal Transport Theory, §§1.1 and 1.3

Depends on

Used by

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

Gabriel Peyré and Marco Cuturi, Computational Optimal Transport, §§2.1–2.3 and 6.1 (standard reference, not scraped)
Filippo Santambrogio, Introduction to Optimal Transport Theory, §§1.1 and 1.3, PDF pp. 3–6 (standard reference, not scraped)