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.

Metric distance-stress multidimensional scaling

Definition

For a symmetric nonnegative zero-diagonal dissimilarity matrix $D \in R^{n \times n}$ , $n \geq 2$ , and a target dimension $q \geq 1$ , use the Euclidean norm $∥ v ∥_{2} = (\sum_{a = 1}^{q} v_{a}^{2})^{1 / 2}$ . The metric stress of centered coordinates $Y = (y_{1}^{T}, \dots, y_{n}^{T})^{T} \in R^{n \times q}$ , with $1_{n}^{T} Y = 0$ , is

$S_{D} (Y) : = \sum_{1 \leq i < j \leq n} (D_{i j} - ∥ y_{i} - y_{j} ∥_{2})^{2} .$

Metric stress MDS is the separate optimization problem of minimizing $S_{D} (Y)$ over such centered $Y$ . It is a distance-based, generally nonconvex objective and does not define the positive-eigenvalue solution used by classical multidimensional scaling; no solver convergence or global-optimizer guarantee is part of this definition.

Definition references

Classical multidimensional scaling

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

Depends on

Classical multidimensional scaling from a finite dissimilarity matrix

Used by

The metric stress objective is not convex even for two points Example

Dependency tree · two levels

2 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

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