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

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A nonzero Gaussian direction forces logarithmic small-ball cost

Statement

Let $W$ be a centered Gaussian random element in a real normed space $B$ . Let $ℓ \in B^{*}$ satisfy $Var (ℓ (W)) = τ^{2} > 0$ , and let $H$ be a Hilbert subspace of $B$ with nonnegative squared norm. For $w_{0} \in B$ , define $ϕ_{w_{0}} (ϵ) = \inf_{\begin{array}{c} h \in H \\ ∥ h - w_{0} ∥_{B} < ϵ \end{array}} {∥ h ∥_{H}^{2}} - \log \Pr (∥ W ∥_{B} < ϵ),$ with $\inf \emptyset = + \infty$ . Then for every $ϵ > 0$ , $ϕ_{w_{0}} (ϵ) \geq \log (1 / ϵ) - C_{ℓ}, C_{ℓ} = \log ⁣ (\frac{2 ∥ ℓ ∥_{B^{*}}}{\sqrt{2 π} τ}) .$

Facts & Assumptions

Given: A centered Gaussian random element, a continuous linear functional with positive variance, and the displayed nonnegative RKHS approximation penalty.

[F1]

Standard normal and normal laws: a centered scalar Gaussian with variance $τ^{2} > 0$ has density bounded above by $(\sqrt{2 π} τ)^{- 1}$ .

[F2]

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

Proof

1.1F1givenalgebra

On the event ${∥ W ∥_{B} < ϵ}$ , continuity and linearity give $∣ ℓ (W) ∣ \leq ∥ ℓ ∥_{B^{*}} ∥ W ∥_{B} < ∥ ℓ ∥_{B^{*}} ϵ$ . The interval $(- ∥ ℓ ∥ ϵ, ∥ ℓ ∥ ϵ)$ has length $2 ∥ ℓ ∥ ϵ$ , so its scalar Gaussian probability is at most its length times the density bound in [F1]. Therefore $\Pr (∥ W ∥_{B} < ϵ) \leq \frac{2 ∥ ℓ ∥_{B^{*}}}{\sqrt{2 π} τ} ϵ .$

2.1F2step 1.1givenalgebra∎

The approximation penalty in the displayed definition is nonnegative. Taking negative logarithms of the bound in step 1.1 and adding this penalty yields $ϕ_{w_{0}} (ϵ) \geq \log (1 / ϵ) - \log (2 ∥ ℓ ∥_{B^{*}} / (\sqrt{2 π} τ))$ , which is the asserted constant.

Source notes

The source defines the Gaussian concentration function as an RKHS approximation cost plus centered small-ball exponent. The one-dimensional projection bound is proved here and does not use a source inequality.

Depends on

Used by

Gaussian-process posterior contraction in density and regression models Theorem

Dependency tree · two levels

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Sources

A. W. van der Vaart and J. H. van Zanten, Rates of contraction of posterior distributions based on Gaussian process priors, equation (1.2) (standard reference, not scraped)
Bas J. K. Kleijn, Aad W. van der Vaart and Harry van Zanten, Lectures on Nonparametric Bayesian Statistics, Definition 11.7 and Theorem 12.1 (standard reference, not scraped)