TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-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.

Kernel ridge residuals certify coefficient and training prediction errors

Statement

Let $K \in R^{n \times n}$ be symmetric with $x^{T} K x \geq 0$ for every $x \in R^{n}$ and $n \geq 1$ , let $y \in R^{n}$ , and let $τ > 0$ . Write $I_{n}$ for the identity matrix. Define $a = (K + τ I_{n})^{- 1} y$ and $u = K a$ . For any vector $b \in R^{n}$ , define the actual residual $r = y - (K + τ I_{n}) b$ . Define $∥ x ∥_{2} = (\sum_{i} x_{i}^{2})^{1 / 2}$ and $∥ M ∥_{2} = \sup_{∥ x ∥_{2} = 1} ∥ M x ∥_{2}$ ; then

$∥ b - a ∥_{2} \leq \frac{∥ r ∥_{2}}{τ}, ∥ K b - u ∥_{2} \leq \frac{∥ K ∥_{2} ∥ r ∥_{2}}{τ} .$

Statement references

Regularized kernel prediction system: The exact coefficient and fitted vectors solve $(K + τ I) a = y$ and $u = K a$ .

Inverse norm bound for a regularized Gram matrix: The shifted Gram inverse has induced norm at most $1 / τ$ .

Facts & Assumptions

Given: The positive-semidefinite matrix $K$ , vector $y$ , scalar $τ > 0$ , exact solution $a$ , and arbitrary approximate coefficient vector $b$ in the Statement.

[F1]

Regularized kernel prediction system: $(K + τ I_{n}) a = y$ and $u = K a$ .

[F2]

Inverse norm bound for a regularized Gram matrix: $K + τ I_{n}$ is invertible and $∥ (K + τ I_{n})^{- 1} ∥_{2} \leq 1 / τ$ .

Proof

1.1givenF1F2algebra

By [F1], $r = y - (K + τ I_{n}) b = (K + τ I_{n}) (a - b)$ . Invertibility in [F2] therefore gives $b - a = - (K + τ I_{n})^{- 1} r$ , so $∥ b - a ∥_{2} \leq ∥ r ∥_{2} / τ$ .

2.1givenF1step 1.1algebra∎

Since $u = K a$ , the training fitted-vector error is $K b - u = K (b - a)$ . For every vector $x$ , the definition of the induced norm gives $∥ K x ∥_{2} \leq ∥ K ∥_{2} ∥ x ∥_{2}$ . Applying this to $b - a$ and using step 1.1 proves $∥ K b - u ∥_{2} \leq ∥ K ∥_{2} ∥ r ∥_{2} / τ$ .

Depends on

Used by

Dependency tree · two levels

4 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

Alex Gittens and Michael W. Mahoney, Revisiting the Nyström Method, §4.1.1 (standard reference, not scraped)
Sheldon Axler, Linear Algebra Done Right, 4th ed., §§7B–7C (standard reference, not scraped)