AlgorithmAlgorithm: 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.

Classical multidimensional scaling from a finite dissimilarity matrix

Definition

Let $n \geq 1$ and let $D \in R^{n \times n}$ be symmetric with nonnegative entries and zero diagonal, and write $D^{\circ 2}$ for the matrix with entries $(D^{\circ 2})_{i j} = D_{i j}^{2}$ . Define $H_{n} = I_{n} - n^{- 1} 1_{n} 1_{n}^{T}$ and the centered matrix

$B : = - \frac{1}{2} H_{n} D^{\circ 2} H_{n} .$

For a requested dimension $q \geq 1$ , classical multidimensional scaling diagonalizes the real symmetric matrix $B$ , retains up to the $q$ largest strictly positive eigenvalues $λ_{j}$ with orthonormal eigenvectors $u_{j}$ , and returns coordinates whose columns are $u_{j} \sqrt{λ_{j}}$ . If fewer than $q$ positive eigenvalues exist, it returns only those positive-eigenvalue coordinates. For non-Euclidean dissimilarities this is a spectral procedure; it does not assert exact recovery of all entries of $D$ .

Definition references

Finite centering matrix

Hastie, Tibshirani, and Friedman, ESL II, §14.8, printed pp. 570–572, and §18.5.2, printed pp. 670–672

Cameron, Linear and Nonlinear Dimensionality Reduction, §3, PDF pp. 4–6

Depends on

Finite centering matrices and centered data

Used by

Dependency tree · one level

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Sources

The Elements of Statistical Learning, second edition, corrected 12th printing (standard reference, not scraped)
Maria Cameron, Linear and Nonlinear Dimensionality Reduction, lecture notes (standard reference, not scraped)