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.

Kernel principal component analysis for a finite positive-semidefinite Gram matrix

Definition

Given $n \geq 1$ data points and a finite symmetric Gram matrix $K \in R^{n \times n}$ satisfying $a^{T} K a \geq 0$ for every $a \in R^{n}$ , define $H_{n} = I_{n} - n^{- 1} 1_{n} 1_{n}^{T}$ and the centered Gram matrix $K_{c} = H_{n} K H_{n}$ . The positive-semidefinite condition is an input hypothesis; no theorem that an arbitrary kernel formula produces such a matrix is assumed. For $1 \leq q \leq n$ , kernel PCA retains the $q$ largest eigenvalues $λ_{1} \geq \dots \geq λ_{n} \geq 0$ of $K_{c}$ and associated orthonormal eigenvectors $u_{1}, \dots, u_{q}$ , returning the score matrix $U_{q} diag (\sqrt{λ_{1}}, \dots, \sqrt{λ_{q}})$ . Eigenvectors inside a tied eigenspace may be chosen arbitrarily.

Definition references

Finite centering matrix

Hastie, Tibshirani, and Friedman, ESL II, §14.5.4, printed pp. 547–550, and §18.5.2, printed pp. 670–672

Cameron, Linear and Nonlinear Dimensionality Reduction, §2.2, PDF pp. 3–4

Depends on

Finite centering matrices and centered data

Used by

Double-centering a finite positive-semidefinite Gram matrix preserves positive semidefiniteness Theorem

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)