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

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Entrywise deviation for sample-centered covariance under directional Gaussian tails

Statement

Let $X_{1}, \dots, X_{n}$ be iid from $P_{q} (τ, c_{n, p})$ of Weak-lq covariance class with directional sub-Gaussian tails, with $n \geq 1$ , $p \geq 2$ , and $τ > 0$ . Let $u > 0$ and let ${\hat{Σ}}^{*}$ be the sample-centered covariance. Then $P ⁣ (\max_{1 \leq j, k \leq p} ∣ {\hat{σ}}_{j k}^{*} - σ_{j k} ∣ \leq 192 τ (\sqrt{\frac{u}{n}} + \frac{u}{n})) \geq 1 - 2 p e^{- u} - 2 p^{2} e^{- 2 u} .$ In particular $u = \log (8 p^{4})$ gives the original probability $1 - p^{- 2}$ . No independence between coordinates of a single observation is assumed.

Facts & Assumptions

Given: Iid vectors $X_{i} \sim P \in P_{q} (τ, c_{n, p})$ , with mean $μ$ and covariance $Σ$ , integers $n \geq 1$ , $p \geq 2$ , and $τ > 0$ .

[F1]

Every unit-direction projection of $X_{i} - μ$ satisfies $P (∣ v^{T} (X_{i} - μ) ∣ > t) \leq e^{- t^{2} / (2 τ)}$ (Weak-lq covariance class with directional sub-Gaussian tails).

[F2]

For two variables with these directional tails, their centered product is sub-exponential with parameters $(1024 τ^{2}, 16 τ)$ (Products of variables with Gaussian tails are sub-exponential).

[F3]

Independent centered sub-exponential sums with total variance proxy $v$ and maximum scale $b$ obey the two-sided Bernstein bound (Sub-exponential concentration for independent sums).

[F4]

The directional tail bound implies a centered sub-Gaussian MGF proxy $64 τ$ (A Gaussian tail bound implies a sub-Gaussian MGF bound).

[F5]

A finite maximum of centered sub-Gaussian variables with common proxy $a^{2}$ has tail at most $2 m e^{- t^{2} / (2 a^{2})}$ (Finite sub-Gaussian maxima and epsilon-net reduction).

[F6]

The sample vectors are iid and mutually independent; applying the coordinate maps $x \mapsto x_{j} - μ_{j}$ preserves independence, so their coordinate samples are independent across $i$ (Identical distribution and IID families, Independent random elements, Measurable coordinatewise functions preserve independence).

[F7]

Expectations of products of independent nonnegative measurable functions factor (Expectations factor over finite products of independent random variables).

[F8]

The covariance entry is $σ_{j k} = E [(X_{j} - μ_{j}) (X_{k} - μ_{k})]$ , and ${\hat{Σ}}^{*} = n^{- 1} \sum_{i} (X_{i} - \overset{ˉ}{X}) (X_{i} - \overset{ˉ}{X})^{T}$ (Moments, variance, and covariance on a probability space, Sample covariance estimators and squared operator loss).

[F9]

Probability is subadditive on finite unions (Basic identities for a probability measure).

Proof

1.1F1F2F6F8given

Put $U_{i} = X_{i} - μ$ . For coordinate unit vectors $e_{j}, e_{k}$ , [F1] gives the required tail bounds for $U_{i j}$ and $U_{i k}$ . Hence [F2] makes $W_{i}^{j k} = U_{i j} U_{i k} - σ_{j k}$ centered sub-exponential with parameters $(1024 τ^{2}, 16 τ)$ . For fixed $(j, k)$ these variables are independent across $i$ by [F6].

1.2F4F5F6F7algebra

By [F4], each $U_{i j}$ is sub-Gaussian with proxy $64 τ$ . Independence across $i$ and [F7] give $E e^{λ n^{- 1} \sum_{i} U_{i j}} \leq \prod_{i} e^{32 τ λ^{2} / n^{2}} = e^{32 τ λ^{2} / n},$ so each ${\overset{ˉ}{U}}_{j} = n^{- 1} \sum_{i} U_{i j}$ has proxy $64 τ / n$ . Apply [F5] at $s = \sqrt{128 τ u / n}$ to obtain $P ⁣ (\max_{j} ∣ {\overset{ˉ}{U}}_{j} ∣ > s) \leq 2 p e^{- u} .$

2.1F3F9step 1.1algebra

Set $r = u / n$ and $t = 64 τ (\sqrt{r} + r)$ . Applying [F3] to $n^{- 1} \sum_{i} W_{i}^{j k}$ gives $P ⁣ (∣ n^{- 1} \sum_{i} W_{i}^{j k} ∣ > t) \leq 2 \exp ⁣ [- \frac{1}{2} \min ⁣ {\frac{n t^{2}}{1024 τ^{2}}, \frac{n t}{16 τ}}] \leq 2 e^{- 2 u},$ because each expression inside the minimum is at least $4 u$ . A union bound over $p^{2}$ ordered pairs gives failure probability at most $2 p^{2} e^{- 2 u}$ .

3.1F8F9step 2.1step 1.2algebra∎

Expanding [F8] gives ${\hat{σ}}_{j k}^{*} = n^{- 1} \sum_{i} U_{i j} U_{i k} - {\overset{ˉ}{U}}_{j} {\overset{ˉ}{U}}_{k}$ . On the intersection of the mean event in step 1.2 and the entrywise event in step 2.1, $∣ {\overset{ˉ}{U}}_{j} {\overset{ˉ}{U}}_{k} ∣ \leq ({\overset{ˉ}{U}}_{j}^{2} + {\overset{ˉ}{U}}_{k}^{2}) / 2 \leq s^{2}$ , so every entry error is at most $64 τ (\sqrt{r} + r) + 128 τ r \leq 192 τ (\sqrt{r} + r) .$ The two failure probabilities sum to at most $2 p e^{- u} + 2 p^{2} e^{- 2 u}$ by [F9]. At $u = \log (8 p^{4})$ this is at most $p^{- 2}$ , proving the original specialization as well.

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Sources

T. Tony Cai and Harrison H. Zhou, Optimal Rates of Convergence for Sparse Covariance Matrix Estimation, Section 4, equations (26)–(27) (standard reference, not scraped)
Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Chapter 4 (standard reference, not scraped)