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

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Influence-curve representers form an affine orthogonal-complement class

Statement

Let $H$ be a real Hilbert space, let $T \subseteq H$ be a closed linear subspace, and let $\dot{ψ} : T \to R^{k}$ be continuous linear with an ambient representer $ϕ_{0} \in H^{k}$ , meaning $\dot{ψ} (s) = ⟨ ϕ_{0}, s ⟩_{H}$ coordinatewise for all $s \in T$ . Assume Countable Choice. Then all ambient representers are exactly

$ϕ_{0} + (T^{⊥})^{k} .$

The unique representer in $T^{k}$ is $ϕ_{c a n} = Π_{T} ϕ_{0}$ ; when $\dot{ψ}$ is a statistical path derivative, this is its canonical gradient. Projection is applied coordinatewise.

Facts & Assumptions

Given: The hypotheses above. Here $T^{⊥} = {h \in H : ⟨ h, s ⟩_{H} = 0 for every s \in T}$ .

[F1]

Hilbert space: a real Hilbert space is a real inner-product space complete in its induced norm.

[F2]

Orthogonal decomposition by a closed subspace: under Countable Choice, a closed subspace $M$ of a real Hilbert space has the decomposition $H = M \oplus M^{⊥}$ .

[F3]

Orthogonality and the orthogonal complement: $M^{⊥}$ consists of the vectors whose inner product with every element of $M$ is zero.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice selects a point from each member of a countable family of nonempty sets; it is the axiom assumed by [F2].

Proof

Given: A real Hilbert space $H$ , a closed linear subspace $T$ , a continuous linear map $\dot{ψ}$ , an ambient representer $ϕ_{0}$ , and Countable Choice.

1.1givenF1F3

If $ϕ \in H^{k}$ is any other representer, then for every $s \in T$ and coordinate $j$ , $⟨ ϕ_{j} - ϕ_{0, j}, s ⟩_{H} = 0$ , so $ϕ_{j} - ϕ_{0, j} \in T^{⊥}$ by [F3]. Thus every representer lies in $ϕ_{0} + (T^{⊥})^{k}$ .

1.2givenF1F2F3A1choose

Apply [F2] coordinatewise to write $ϕ_{0} = Π_{T} ϕ_{0} + h$ with $Π_{T} ϕ_{0} \in T^{k}$ and $h \in (T^{⊥})^{k}$ . Countable Choice is used only in this projection step through [F2] and [A1].

2.1givenF1F3step 1.1algebra

Conversely, if $h \in (T^{⊥})^{k}$ , then $⟨ ϕ_{0} + h, s ⟩_{H} = ⟨ ϕ_{0}, s ⟩_{H} + ⟨ h, s ⟩_{H} = \dot{ψ} (s)$ for every $s \in T$ , so $ϕ_{0} + h$ is a representer. Hence the full representer set is exactly the stated affine class.

3.1step 1.1step 1.2step 2.1F1F3algebra∎

By step 1.2, $Π_{T} ϕ_{0}$ is a representer. If two representers lie in $T^{k}$ , step 1.1 places their difference in $(T^{⊥})^{k}$ as well; each coordinate then lies in $T \cap T^{⊥}$ and has squared norm zero by positive definiteness in [F1], so the difference is zero. This proves existence and uniqueness of the canonical representer.

Remarks

The theorem is a Hilbert-space statement; its only choice use is Countable Choice for the projection decomposition.

Depends on

Used by

Dependency tree · two levels

16 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

A. W. van der Vaart, Semiparametric Statistics, Lecture Notes in Mathematics 1781, Lectures 1–3 (Springer, 2002) (standard reference, not scraped)
Edward H. Kennedy, Semiparametric theory, arXiv:1709.06418v1 (2017) (standard reference, not scraped)