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

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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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High-probability operator error for thresholded weak-lq covariance

Statement

Let $X_{1}, \dots, X_{n}$ be iid (Identical distribution and IID families) from $P_{q} (τ, c_{n, p})$ of Weak-lq covariance class with directional sub-Gaussian tails, with $p \geq 2$ , $n \geq 1$ , $0 \leq q < 1$ , $τ > 0$ , and $\log p \leq n$ . Use Entrywise-thresholded sample covariance with threshold $λ = 8192 τ \sqrt{\frac{\log p}{n}} .$ Then for a finite constant $C_{q}$ depending only on $q$ , $P ⁣ (∥ \hat{Σ} - Σ ∥_{o p} \leq C_{q} [c_{n, p} {(τ \sqrt{\frac{\log p}{n}})}^{1 - q} + τ \sqrt{\frac{\log p}{n}}]) \geq 1 - p^{- 2} .$ This is a high-probability bound; it does not assert an expected-risk or minimax lower bound.

Facts & Assumptions

Given: The iid law, dimensions, tail parameter and weak-lq covariance class in the Statement, together with the thresholded estimator at the displayed $λ$ .

[F1]

On the entrywise event with error at most $λ / 2$ , thresholding gives operator error at most $C_{q} (c_{n, p} λ^{1 - q} + λ)$ (Weak-lq thresholding error from an entrywise deviation event).

[F2]

For $u = \log (8 p^{4})$ , the sample-centered covariance obeys $\max_{j, k} ∣ {\hat{σ}}_{j k}^{*} - σ_{j k} ∣ \leq 192 τ (\sqrt{u / n} + u / n)$ with probability at least $1 - p^{- 2}$ (Entrywise deviation for sample-centered covariance under directional Gaussian tails).

[F3]

The algorithm leaves the diagonal unchanged and thresholds each off-diagonal entry at $λ$ (Entrywise-thresholded sample covariance).

Proof

1.1F2givenalgebra

Put $m = \log p / n$ and $r = u / n$ . Since $p \geq 2$ , $\log 8 = 3 \log 2 \leq 3 \log p$ , so $u = \log 8 + 4 \log p \leq 7 \log p$ and $r \leq 7 m \leq 7$ . Thus $\sqrt{r} + r \leq (1 + \sqrt{7}) \sqrt{r} < 4 \sqrt{r} < 12 \sqrt{m}$ . By [F2], on an event of probability at least $1 - p^{- 2}$ the entrywise error is less than $2304 τ \sqrt{m} < 4096 τ \sqrt{m} = λ / 2$ .

2.1F1F3step 1.1algebra∎

On the event in step 1.1, [F1] applies to the thresholded estimator of [F3] and yields $∥ \hat{Σ} - Σ ∥_{o p} \leq C_{q} (c_{n, p} λ^{1 - q} + λ) .$ Substituting $λ = 8192 τ \sqrt{m}$ and absorbing the fixed factor $8192^{1 - q}$ into a finite constant depending only on $q$ gives the stated bound. The event probability is at least $1 - p^{- 2}$ .

Depends on

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Sources

T. Tony Cai and Harrison H. Zhou, Optimal Rates of Convergence for Sparse Covariance Matrix Estimation, Section 4, equations (28)–(29) (standard reference, not scraped)
Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Chapter 4 (standard reference, not scraped)