ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

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A dimension-dependent lower bound for empirical W1 error

Example

Let $M \in N$ with $M \geq 1$ , set $V_{M} = {0, 1}^{M} \subset R^{M}$ , and let $μ_{M}$ be uniform on its $2^{M}$ vertices. For any empirical measure $ν_{N} = N^{- 1} \sum_{j = 1}^{N} δ_{Y_{j}}$ obtained from $N$ samples $Y_{j} \in V_{M}$ , the 1-Wasserstein error has the deterministic lower bound $W_{1} (μ_{M}, ν_{N}) \geq 1 - \frac{N}{2^{M}} .$ Thus no dimension-free bound $W_{1} (μ, μ_{N}) \leq C N^{- 1 / 2}$ can hold uniformly over all dimensions and all empirical samples.

Facts & Assumptions

Given: The uniform probability on $V_{M}$ , an integer $N \geq 1$ , and any list of sample vertices $Y_{1}, \dots, Y_{N} \in V_{M}$ .

[F1]

$W_{1}$ is the infimum of mean Euclidean transport costs over probability couplings (The coupling definition of the 1-Wasserstein distance).

Verification

Given: The data in the Example.

1.1givenalgebra

Let $S = {Y_{1}, \dots, Y_{N}}$ be the empirical support. It contains at most $N$ vertices, so $μ_{M} (V_{M} ∖ S) \geq 1 - N / 2^{M}$ . Distinct vertices of $V_{M}$ have Euclidean distance at least one.

2.1F1step 1.1algebra∎

In any coupling of $μ_{M}$ to $ν_{N}$ , every point of $V_{M} ∖ S$ must send its mass to a point of $S$ , at cost at least one per unit mass. Hence every coupling costs at least $1 - N / 2^{M}$ , and taking the infimum in [F1] proves the bound. If $2^{M} \geq 2 N$ , it is at least $1 / 2$ ; choosing $N$ arbitrarily large and then such an $M$ rules out a uniform dimension-free $C N^{- 1 / 2}$ bound.

Depends on

The coupling definition of the 1-Wasserstein distance

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

François Golse, Mean-Field Limits in Statistical Dynamics, Lecture 1 (standard reference, not scraped)