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.

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A coefficient below the orthonormal Lasso threshold is set to zero

Example

For the one-dimensional fixed design $n = p = 1$ , $X = [1]$ , response $y = β^{*} = 1 / 4$ , and Lasso penalty $λ = 1 / 2$ , the true support is ${1}$ but the unique Lasso minimizer is $0$ . This is a deterministic example with no observation noise; it asserts no random-design probability result.

Example references

The Elements of Statistical Learning, §3.4.1

Rigollet and Hütter, §2.3

Facts & Assumptions

Given: $n = p = 1$ , $X = [1]$ , $y = β^{*} = 1 / 4$ , no noise, and $λ = 1 / 2$ .

[F1]

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

Verification

1.1F1givenalgebra

Here $X^{T} X = 1$ , $n λ = 1 / 2$ , and $∣ X^{T} y ∣ = 1 / 4 < 1 / 2$ . Therefore $S_{1 / 2} (1 / 4) = 0$ by [F1].

2.1step 1.1givenalgebra∎

The true coefficient is nonzero, so its support is ${1}$ , whereas the estimated coefficient is zero and its support is empty. Thus this choice of penalty misses the support when the coefficient is below the threshold.

Depends on

Lasso on an orthonormal design is coordinatewise soft-thresholding

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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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)