TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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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Finite-grid GCV is well-defined and minimizes its supplied score

Statement

For every admissible input to Finite-grid generalized cross-validation for Tikhonov regularization, all GCV scores are finite real numbers, the algorithm terminates, and its output has the least score on the supplied grid, with ties resolved by the least index. This is a score-minimization statement only; it gives no bound on solution error or universal statistical optimality.

Facts & Assumptions

Given: A real matrix $A \in R^{m \times n}$ , $m \geq 1$ , data $b \in R^{m}$ , and a nonempty finite increasing positive grid, as in Finite-grid generalized cross-validation for Tikhonov regularization.

[F1]

Finite-grid generalized cross-validation for Tikhonov regularization: The algorithm computes the Tikhonov hat matrix and the stated GCV score at each grid point, then returns a least-score parameter with least-index tie breaking.

[F2]

The ridge leave-one-out identity yields the generalized cross-validation score: For every $λ > 0$ , $1 - tr (H_{λ}) / m > 0$ and the score equals $∥ (I - H_{λ}) b ∥^{2} / [m (1 - tr (H_{λ}) / m)^{2}]$ .

Proof

1.1givenF1F2algebra

Each $λ_{j} > 0$ makes $A^{T} A + λ_{j} I$ positive definite, so all matrix inverses in [F1] exist. By [F2], the denominator of each score is strictly positive; the numerator is a finite squared norm. Thus each score is a finite real number. The procedure performs finitely many operations for finitely many grid entries and terminates.

2.1F1step 1.1algebra∎

A nonempty finite list of real scores has a minimum, and a nonempty finite set of indices attaining the minimum has a least element. By the selection rule in [F1], the algorithm returns that index and hence a least-score grid parameter. No part of this finite comparison identifies the score with the unknown solution error.

Depends on

Used by

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Sources

Per Christian Hansen, Introduction to Inverse Problems, Chapter 5: Regularization Parameter Choice (standard reference, not scraped)
Guido Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)