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Squared-norm concentration for real and normalized complex Gaussians

Statement

Assume Countable Choice for the finite-dimensional Gaussian probability law. Let $m \geq 1$ , and let $g_{1}, \dots, g_{m}$ be independent standard real Gaussian scalars or independent normalized complex Gaussian scalars, with $E ∣ g_{j} ∣^{2} = 1$ . For $0 < t < 1$ ,

\Pr {∣ \frac{1}{m} \sum_{j = 1}^{m} ∣ g_{j} ∣^{2} - 1 ∣ \geq t} \leq 2 e^{- m t^{2} / 8} .

Facts & Assumptions

Given: Countable Choice for the finite Gaussian law, and a finite family of independent standard Gaussian scalars in one of the two declared fields and a parameter $0 < t < 1$ .

[F1]

A standard real Gaussian scalar has density $(2 π)^{- 1 / 2} e^{- x^{2} / 2}$ ; a normalized complex one has density $π^{- 1} e^{- ∣ z ∣^{2}}$ and mean squared modulus one (Standard real and normalized complex Gaussian matrix laws). The real and complex densities each have total mass one, zero mean and second moment one under exactly the stated Countable Choice; finite products have unit mass and independent coordinates by Normalization and second moments of the Gaussian matrix laws.

[F2]

Countable Choice, written ${A C}_{ω}$ , is used in the repository's construction of the finite-dimensional Lebesgue probability measures for these densities (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F3]

Under Countable Choice, invertible real linear maps scale Lebesgue measure by the absolute determinant (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not). The corresponding nonnegative-integral identity follows first for indicators, then simple functions and their defining suprema.

[F4]

For a nonnegative measurable random variable $X$ and $a > 0$ , $\Pr {X \geq a} \leq E X / a$ (Markov's inequality for random variables).

[F5]

$\log x = \int_{1}^{x} d u / u$ for $x > 0$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t).

[F6]

The probability of a finite union of events is at most the sum of their probabilities (The finite union bound).

[A1]

We use only Countable Choice for those finite Gaussian integrals; no unrestricted Axiom of Choice is assumed.

Proof

Given: The hypotheses in the Statement.

1.1F1F2F3A1algebra

For real $Z$ and $θ < 1 / 2$ , put $c = 1 - 2 θ > 0$ . By [F3], the substitution $u = \sqrt{c} x$ gives $E e^{θ Z^{2}} = c^{- 1 / 2} (2 π)^{- 1 / 2} \int_{R} e^{- u^{2} / 2} d u = c^{- 1 / 2}$ . The prefactored integral is one since [F1] is a probability density. For normalized complex $G$ and $θ < 1$ , put $d = 1 - θ > 0$ . On $C ≃ R^{2}$ , the map $u = \sqrt{d} z$ has real determinant $d$ , so [F3] gives $E e^{θ ∣ G ∣^{2}} = d^{- 1} π^{- 1} \int_{R^{2}} e^{- ∣ u ∣^{2}} d u = d^{- 1}$ . Again the prefactored integral is one by the normalized density [F1]. These identities include the finite integrability ranges needed below.

2.1F1F4F5step 1.1algebra

For the real case, [F4] applied to $X = e^{λ \sum_{j} g_{j}^{2}}$ at the positive threshold $a = e^{λ m (1 + t)}$ , with $λ = t / (2 (1 + t))$ , and independence together with the first formula of step 1.1 give $\Pr {m^{- 1} \sum_{j} g_{j}^{2} \geq 1 + t} \leq \exp [- \frac{m}{2} (t - \log (1 + t))]$ . For the complex case use $X = e^{λ \sum_{j} ∣ g_{j} ∣^{2}}$ , the same threshold, $λ = t / (1 + t)$ , and the second formula to obtain the stronger bound $\exp [- m (t - \log (1 + t))]$ . Both positive parameters lie in the finite mgf ranges of step 1.1. By [F5], $t - \log (1 + t) = \int_{0}^{t} u / (1 + u) d u \geq t^{2} / 4$ , so both upper tails are at most $e^{- m t^{2} / 8}$ .

2.2F1F4F5step 1.1algebra

For the real lower tail, apply [F4] to $X = e^{- λ \sum_{j} g_{j}^{2}}$ at threshold $a = e^{- λ m (1 - t)} > 0$ with $λ = t / (2 (1 - t))$ ; for the complex lower tail use $X = e^{- λ \sum_{j} ∣ g_{j} ∣^{2}}$ , the same threshold, and $λ = t / (1 - t)$ . The event $X \geq a$ is precisely the lower-tail event because the exponential is decreasing in the squared norm sum. The mgf arguments $- λ$ lie in the finite ranges of step 1.1, and independence again factors the expectations. Step 1.1 gives respective bounds $\exp [- \frac{m}{2} (- t - \log (1 - t))]$ and $\exp [- m (- t - \log (1 - t))]$ . By [F5], $- t - \log (1 - t) = \int_{0}^{t} u / (1 - u) d u \geq t^{2} / 2$ , so each is at most $e^{- m t^{2} / 8}$ .

3.1step 2.1step 2.2F6algebra∎

The event in the Statement is the union of the upper and lower tail events bounded in steps 2.1 and 2.2. The finite union bound [F6] therefore gives the claimed sum bound.

Depends on

Used by

Gaussian sketches embed a fixed subspace at the additive dimension rate Theorem

Dependency tree · two levels

57 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

David P. Woodruff, Sketching as a Tool for Numerical Linear Algebra, §2.1, Theorem 4 and §2.2, Lemma 18 (standard reference, not scraped)
Michael W. Mahoney, Randomized Algorithms for Matrices and Data, §3.1 (standard reference, not scraped)