ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

An orthonormal design makes the Lasso threshold visible coordinate by coordinate

Example

Let $n = p = 2$ , $X = I_{2}$ , $y = (3, - 1)^{T}$ , and use Lasso penalty $λ = 1 / 2$ . Under the empirical loss $∥ y - X β ∥_{2}^{2} / (2 n)$ , the unique Lasso minimizer is $(2, 0)^{T}$ because the threshold is $n λ = 1$ . OLS is $(3, - 1)^{T}$ , and ridge with parameter $1$ is $(3 / 2, - 1 / 2)^{T}$ .

Example references

The Elements of Statistical Learning, §3.4.1

Rigollet and Hütter, §2.3

Facts & Assumptions

Given: $n = p = 2$ , $X = I_{2}$ , $y = (3, - 1)^{T}$ , Lasso penalty $1 / 2$ , and ridge parameter $1$ .

[F1]

On an orthonormal design, the Lasso minimizer is coordinatewise soft-thresholding at $n λ$ (Lasso on an orthonormal design is coordinatewise soft-thresholding).

[F2]

OLS and ridge have the formulas stated in The full ordinary least-squares solution set and the unique ridge solution.

Verification

1.1F2givenalgebra

Since $X^{T} X = I_{2}$ , OLS gives $X^{T} y = (3, - 1)^{T}$ , and the ridge formula with parameter $1$ gives $(I_{2} + I_{2})^{- 1} y = (3 / 2, - 1 / 2)^{T}$ .

2.1F1givenalgebra∎

The Lasso threshold is $n λ = 2 (1 / 2) = 1$ . Applying [F1] gives $S_{1} (3) = 2$ and $S_{1} (- 1) = 0$ , hence $(2, 0)^{T}$ . The equality case $∣ y_{2} ∣ = n λ$ is assigned zero by the stated soft-threshold formula.

Depends on

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Sources

Hastie, Tibshirani, and Friedman, The Elements of Statistical Learning, 2nd ed., corrected 12th printing, §3.4.1 (standard reference, not scraped)
Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, §2.3 (standard reference, not scraped)