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Gaussian trace/norm estimate attributed to Slepian comparison

PDF p. 34: Proof of Proposition 6.3, citation to [7, Lemma 2.2]

Statement

Assume the Axiom of Choice. Let $n \geq 1$ and $ℓ \geq 1$ be integers, and let $G$ be the symmetric Gaussian comparison matrix of Gaussian comparison matrix in the moment argument: its independent triangular entries, including the diagonal, all have law $N (0, 1)$ . Use the operator norm of Loewner order, Hermitian inner product, and operator norm and write $Tr (B) = \sum_{i = 1}^{n} B_{i i}$ . Then

$(E Tr (G^{2 ℓ}))^{1 / (2 ℓ)} \leq n^{1 / (2 ℓ)} (E ∥ G ∥^{2 ℓ})^{1 / (2 ℓ)} \leq n^{1 / (2 ℓ)} (2 \sqrt{n} + 2 \sqrt{2 ℓ}) .$

Facts & Assumptions

[F1]

The triangular realization $G = T (Z)$ , where $Z$ has $N = n (n + 1) / 2$ independent standard normal coordinates, has every even norm moment. The extreme values $λ_{+} = \sup_{∥ v ∥_{2} = 1} v^{T} G v$ and $λ_{-} = - \inf_{∥ v ∥_{2} = 1} v^{T} G v$ have common mean $0 \leq m \leq 2 \sqrt{n}$ . Their coordinate functions are $\sqrt{2}$ -Lipschitz, and $∥ G ∥ \leq m + W$ for $W = \max (∣ λ_{+} - m ∣, ∣ λ_{-} - m ∣)$ . Triangular Gaussian coordinates: metric and moment control, Expected extreme eigenvalues from finite Gaussian comparison, Extreme-value Lipschitz bounds and centered norm reduction.

[F2]

For every integer $ℓ \geq 2$ , $(E ∥ G ∥^{2 ℓ})^{1 / (2 ℓ)} \leq 2 \sqrt{n} + 2 \sqrt{2 ℓ}$ for precisely this variance-one diagonal convention. An independent symmetric-Gaussian norm moment input.

[F3]

For every integer $ℓ \geq 1$ , the even-trace moment is at most the corresponding norm moment multiplied by $n^{1 / (2 ℓ)}$ . Even trace moments transfer from the operator norm.

[F4]

Minkowski gives $∥ X + Y ∥_{2} \leq ∥ X ∥_{2} + ∥ Y ∥_{2}$ on a probability space, and the nonnegative integral is monotone, homogeneous and additive. Minkowski's inequality for integrals, including p = ∞, Monotonicity and nonnegative homogeneity of the nonnegative integral, Additivity of the nonnegative Lebesgue integral.

[F5]

The imported Gaussian Poincaré inequality in External input bounds the variance of any globally $L$ -Lipschitz real function of a standard Gaussian vector by $L^{2}$ .

[F6]

Under Choice, $N \geq 1$ mutually independent standard normal coordinates, viewed as one-dimensional normal vectors, stack to a Borel random vector with law $N_{N} (0, I_{N})$ , the standard Gaussian law $γ_{N}$ . The dimension-one multivariate convention agrees with the scalar normal law. A finite stack of independent Gaussian vectors is jointly Gaussian, Multivariate normal law, including singular covariance.

[F7]

Under Countable Choice, continuous maps between positive-dimensional Euclidean spaces are Borel measurable. Expectations of nonnegative or integrable Borel functions of a random element equal their integrals against its law. Continuous functions on Euclidean spaces are Borel measurable, Change of variables for expectation.

External input

For the standard Gaussian probability measure $γ_{N}$ on $R^{N}$ , Ledoux, Section 1, Theorem 1, equation (3), states that every locally Lipschitz $f \in L^{2} (γ_{N})$ satisfies ${Var}_{γ_{N}} (f) \leq \int ∣ \nabla f ∣^{2} d γ_{N}$ . In particular a globally $L$ -Lipschitz function has ${Var}_{γ_{N}} (f) \leq L^{2}$ , since at its almost-everywhere differentiability points every directional derivative is bounded by $L$ times the direction's Euclidean norm. This is an explicit external input, rather than a locally reconstructed concentration theorem.

Proof

Given: The objects and hypotheses in the Statement.

Goal: Prove the full trace/norm estimate, including $ℓ = 1$ .

1.1F1F4F5F6F7given

Use the independent upper-triangular coordinates from the comparator construction as $Z$ . Reflection to a lower-triangular convention only relabels these same coordinates; the diagonal variance stays one. In [F6], take $N = n (n + 1) / 2 \geq 1$ blocks of dimension one, each with mean zero and covariance $[1]$ . Thus $Z$ has law $N_{N} (0, I_{N}) = γ_{N}$ . Put $f_{\pm} (z) = λ_{\pm} (T (z))$ . By [F1] these functions are $\sqrt{2}$ -Lipschitz, hence continuous. Choice implies Countable Choice, so [F7] makes them Borel measurable and transfers their first and second moments to $γ_{N}$ . Their absolute values are at most $∥ T (z) ∥$ , as proved in the centered-extreme reduction; therefore [F1] gives $f_{\pm} \in L^{2} (γ_{N})$ with common integral $m$ . Apply [F5] with $L = \sqrt{2}$ , and use [F7] for the nonnegative Borel functions $(f_{\pm} - m)^{2}$ . This gives $E (λ_{+} - m)^{2} \leq 2$ and $E (λ_{-} - m)^{2} \leq 2$ .

2.1F1F4F7step 1.1algebra

Pointwise, $W^{2} \leq (λ_{+} - m)^{2} + (λ_{-} - m)^{2}$ . All these functions are nonnegative and measurable by [F1] and [F7]. Monotonicity and additivity in [F4], followed by step 1.1, give $E W^{2} \leq E (λ_{+} - m)^{2} + E (λ_{-} - m)^{2} \leq 4$ . The centered reduction and Minkowski therefore imply $(E ∥ G ∥^{2})^{1 / 2} \leq ∥ m + W ∥_{2} \leq m + ∥ W ∥_{2} \leq 2 \sqrt{n} + 2 \leq 2 \sqrt{n} + 2 \sqrt{2}$ . This proves the requested norm bound for $ℓ = 1$ , with the stronger additive constant $2$ .

3.1F2F3step 2.1algebra∎

For integers $ℓ \geq 2$ , apply [F2]. Together with step 2.1 this proves the norm bound for every stated $ℓ$ . Apply [F3] for the same $ℓ$ and multiply the norm bound by the positive factor $n^{1 / (2 ℓ)}$ to obtain the full displayed chain.

Source notes

The original article's PDF p. 34 attributes the displayed chain to Slepian's lemma and Bandeira–van Handel, Lemma 2.2. That lemma states its norm bound for $p \geq 2$ , with the same variance-one diagonal convention. The supplier An independent symmetric-Gaussian norm moment input provides that range; the local argument above supplies $ℓ = 1$ by Gaussian Poincaré and the extreme-value reduction.

Depends on

Used by · 1

Expected-norm estimate from the corrected moment comparison

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Michel Ledoux, The Gaussian Poincaré inequality, The Gaussian Blog (v1 February 2024) · Section 1, Theorem 1, equation (3), printed/PDF p. 3

For every locally Lipschitz f: R^N -> R in L^2(gamma_N), Var_gamma_N(f) <= integral |gradient f|^2 dgamma_N, where gamma_N is the standard Gaussian probability measure. In particular, a globally L-Lipschitz function has variance at most L^2: its almost-everywhere gradient has norm at most L.

Hypotheses: Finite N>=1; standard Gaussian measure on R^N with identity covariance; locally Lipschitz real f with finite second moment. The application uses globally sqrt(2)-Lipschitz extreme-eigenvalue functions of independent variance-one triangular coordinates.

Reading limits: Read the introduction’s Gaussian density and variance convention, Section 1 Theorem 1 (3), and Section 2’s product-measure statement. Only the finite-dimensional Poincaré inequality and its Lipschitz consequence are imported; its analytic proof is not reconstructed.

Afonso S. Bandeira and Ramon van Handel, Sharp nonasymptotic bounds on the norm of random matrices with independent entries · Lemma 2.2, printed/PDF p. 6; proof continues on p. 7; matching triangular convention in Proposition 2.1 on p. 6

Lemma 2.2 states that if Y_r is the r by r symmetric matrix whose entries on and below the diagonal are independent N(0,1) random variables, then (E||Y_r||^(2p))^(1/(2p)) <= 2 sqrt(r) + 2 sqrt(2p) for every p>=2.

Hypotheses: Positive integer dimension r; real symmetric matrix; independent standard normal entries indexed by i>=j, including variance-one diagonal entries; mirrored off-diagonal entries coincide. The lemma states p>=2; this item uses its range only for integer p=ell>=2. Transposing the triangular indexing does not change the matrix law.

Reading limits: Read the extracted PDF text of Lemma 2.2 and its proof on pp. 6–7, together with Proposition 2.1 on p. 6 to check the triangular convention. This reference directly supports the source-note assertion about the lemma’s statement and range. The Gaussian-process and concentration theorems cited in that proof were not independently read or reconstructed; the local argument and its existing suppliers are retained.

Sources

Expander graphs are globally synchronizing — Abdalla, Bandeira, Kassabov, Souza, Strogatz, Townsend (2026) · PDF p. 34, Proof of Proposition 6.3, citation to [7, Lemma 2.2]
Michel Ledoux, The Gaussian Poincaré inequality, The Gaussian Blog (v1 February 2024) · Section 1, Theorem 1, equation (3), printed/PDF p. 3
Afonso S. Bandeira and Ramon van Handel, Sharp nonasymptotic bounds on the norm of random matrices with independent entries · Lemma 2.2, printed/PDF p. 6; proof continues on p. 7; matching triangular convention in Proposition 2.1 on p. 6