ExampleConstruction: AI-generatedVerification: 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 systems with singular and zero Gram matrices

Example

For $τ > 0$ , the singular matrix $K = diag (1, 0)$ and response $y = (p, q)^{T}$ give coefficient vector $a = ((1 + τ)^{- 1} p, τ^{- 1} q)^{T}$ and training fitted vector $u = ((1 + τ)^{- 1} p, 0)^{T}$ . For the zero matrix $K = 0$ of any dimension $n \geq 1$ and any response $y \in R^{n}$ , the coefficient vector is $a = y / τ$ and the fitted vector is $u = 0$ .

Example references

Regularized kernel prediction system: The system is $a = (K + τ I)^{- 1} y$ and $u = K a$ .

These vectors give the unique minimizer whenever the Gram matrix comes from the stated real RKHS kernel.

Facts & Assumptions

Given: $τ > 0$ , $p, q \in R$ , and the singular and zero matrices and responses in the Example.

[A1]

The matrices and responses are exactly those in the Example.

[F1]

Regularized kernel prediction system: the coefficient solves $(K + τ I_{n}) a = y$ and the fitted vector is $u = K a$ .

[F2]

Inverse norm bound for a regularized Gram matrix: if $K ⪰ 0$ and $τ > 0$ , then $K + τ I_{n}$ is invertible.

[F3]

The kernel ridge system uniquely minimizes the mean-square RKHS objective: for a real RKHS kernel and $λ = τ / n > 0$ , the same coefficient vector yields the unique minimizer of the mean-square RKHS objective.

Verification

Given: The matrices and responses in [A1].

1.1givenA1F1F2algebra

For $K = diag (1, 0)$ , $K + τ I_{2} = diag (1 + τ, τ)$ , whose inverse is $diag ((1 + τ)^{- 1}, τ^{- 1})$ . Multiplication by $(p, q)^{T}$ gives the displayed coefficient vector, and multiplication by $K$ gives $u = ((1 + τ)^{- 1} p, 0)^{T}$ .

2.1givenA1F1F2F3algebra∎

For $K = 0$ , $K + τ I_{n} = τ I_{n}$ has inverse $τ^{- 1} I_{n}$ , so $a = y / τ$ and $u = K a = 0$ for every $n \geq 1$ . Both matrices are positive semidefinite, and [F3] identifies their finite systems with the stated RKHS minimizers when the kernel hypotheses hold.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, Chapter 6 §6.2.2 (standard reference, not scraped)