ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The metric stress objective is not convex even for two points

Example

Let two objects have dissimilarity $D_{12} = 1$ and use one-dimensional centered coordinates $Y (t) = (t, - t)$ , $t \in R$ . The metric stress is $f (t) = (1 - 2 ∣ t ∣)^{2}$ , which is not convex: $f (0) = 1$ while $(f (- 1 / 4) + f (1 / 4)) / 2 = 1 / 4$ .

Example references

Hastie, Tibshirani, and Friedman, ESL II, §14.8, printed pp. 570–571

Facts & Assumptions

Given: Two objects with dissimilarity $D_{12} = 1$ , target dimension one, and the centered coordinates $Y (t) = (t, - t)$ .

[F1]

Metric stress is $S_{D} (Y) = \sum_{i < j} (D_{i j} - ∥ y_{i} - y_{j} ∥_{2})^{2}$ on centered coordinates (Metric distance-stress multidimensional scaling).

Verification

1.1givenF1algebra

For $Y (t) = (t, - t)$ , the sole distance is $∣ t - (- t) ∣ = 2 ∣ t ∣$ , so [F1] gives $f (t) = (1 - 2 ∣ t ∣)^{2}$ .

2.1step 1.1algebra∎

A convex function must satisfy the midpoint inequality $f (0) \leq \frac{1}{2} f (- a) + \frac{1}{2} f (a)$ . At $a = 1 / 4$ , $f (0) = 1$ and $f (- a) = f (a) = (1 - 1 / 2)^{2} = 1 / 4$ , so the midpoint inequality fails. Hence this valid stress objective is nonconvex.

Depends on

Metric distance-stress multidimensional scaling

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

The Elements of Statistical Learning, second edition, corrected 12th printing (standard reference, not scraped)