ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Two-point finite-metric transport

Example

Let $X = {x_{1}, x_{2}}$ have metric distance $d (x_{1}, x_{2}) = δ > 0$ . For $a, b \in [0, 1]$ , let $p = (a, 1 - a)$ and $q = (b, 1 - b)$ . Then $W_{1} (p, q) = δ ∣ a - b ∣$ . If $a \geq b$ , one attaining coupling is

π = (\begin{array}{cc} b & a - b \\ 0 & 1 - a \end{array})

and one attaining normalized potential is $f = (0, - δ)$ . If $b > a$ , one attaining coupling and potential are

π = (\begin{array}{cc} a & 0 \\ b - a & 1 - b \end{array}), f = (0, δ) .

Example references

Peyré and Cuturi, Computational Optimal Transport, §6.1

Santambrogio, Introduction to Optimal Transport Theory, §1.3

Facts & Assumptions

Given: The two-point metric space, probability vectors, and parameter bounds in the Example.

[F1]

A coupling is a nonnegative matrix with the stated row and column sums, and $W_{1}$ is its minimum transport cost (Couplings of finite probability vectors, The finite-metric 1-Wasserstein transport cost).

[F2]

The minimum coupling cost equals the maximum over normalized vectors satisfying $∣ f_{i} - f_{j} ∣ \leq d (x_{i}, x_{j})$ ; both extrema are attained (Finite-metric transport duality and complementary slackness).

Verification

1.1givenF1constructalgebra

If $a \geq b$ , all entries of the displayed matrix are nonnegative, its row sums are $b + (a - b) = a$ and $1 - a$ , and its column sums are $b$ and $(a - b) + (1 - a) = 1 - b$ . Its cost is $δ (a - b)$ . If $b > a$ , the second displayed matrix has nonnegative entries, row sums $a$ and $(b - a) + (1 - b) = 1 - a$ , column sums $a + (b - a) = b$ and $1 - b$ , and cost $δ (b - a)$ . Thus in either case it is a feasible coupling of cost $δ ∣ a - b ∣$ .

1.2givenF1algebra

If $a \geq b$ , the vector $(0, - δ)$ is normalized and satisfies the Lipschitz constraint with equality; its objective is $(b - a) (- δ) = δ (a - b)$ . If $b > a$ , the vector $(0, δ)$ is normalized and feasible, and its objective is $(b - a) δ = δ (b - a)$ . Thus the displayed potential has value $δ ∣ a - b ∣$ in either case.

2.1step 1.1step 1.2F2algebra∎

By [F2], the optimal primal and dual values coincide. The feasible plan in step 1.1 gives an optimal primal value at most $δ ∣ a - b ∣$ , while the feasible potential in step 1.2 gives an optimal dual value at least that number. Equality of the optima forces both displayed values to be optimal and gives $W_{1} (p, q) = δ ∣ a - b ∣$ .

Depends on

Used by

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Sources

Gabriel Peyré and Marco Cuturi, Computational Optimal Transport, §6.1 (standard reference, not scraped)
Filippo Santambrogio, Introduction to Optimal Transport Theory, §1.3, PDF pp. 5–6 (standard reference, not scraped)