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

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Polynomial convolutions force activation polynomiality almost everywhere

Statement

Assume Countable Choice. Let $σ \in M$ . If every convolution

$g_{φ} (x) = \int_{R} σ (x - t) φ (t) d t, φ \in C_{c}^{\infty} (R), supp φ \subseteq [- 1, 1],$

is a polynomial, then $σ$ agrees almost everywhere with a polynomial. Equivalently, if $σ$ is not polynomial almost everywhere, at least one such test function has a nonpolynomial convolution.

Statement references

Leshno et al., proof of Theorem 1, distributional polynomiality step

Dyatlov, Lecture Notes for 18.155, mollifier approximation

Facts & Assumptions

Given: Countable Choice and a measurable, locally essentially bounded activation $σ \in M$ .

[F1]

Such an activation is locally integrable, and its regular distribution is defined by integration against test functions (Locally essentially bounded activations with null discontinuity closure, Regular distribution from a locally integrable function).

[F2]

If every convolution with a real-valued test function supported in $[- 1, 1]$ is polynomial, their degrees have a common finite bound (Polynomial activation convolutions have a common degree bound).

[F3]

Unit-mass mollifications converge locally in distribution pairings under Countable Choice (Mollifier approximation in distributions, The mollifier family generated by a unit-mass smooth bump).

[F4]

A distribution whose $(N + 1)$ st derivative vanishes is a polynomial distribution of degree at most $N$ (A distribution with a vanishing derivative is a polynomial).

[F5]

The map from locally integrable functions modulo almost-everywhere equality to regular distributions is injective under Countable Choice (Locally integrable functions embed in distributions).

[F6]

The distributional derivative of a polynomial's regular distribution is its classical derivative, so the derivative of order $N + 1$ of a degree-at-most- $N$ polynomial vanishes (The formal derivative of a polynomial, Distributional differentiation is continuous and commutes).

[F7]

A nonnegative smooth cutoff supported in $(- 1, 1)$ and positive near zero can be normalized by its finite positive integral to obtain a unit-mass bump supported in $[- 1, 1]$ (Test function cutoffs and euclidean localization).

[F8]

Distributional derivatives are defined by $⟨ D^{m} T, ψ ⟩ = (- 1)^{m} ⟨ T, ψ^{(m)} ⟩$ (Distributional derivative).

[F9]

Since $σ$ is real-valued and $T_{σ} (ψ) = \int σ ψ$ , its regular distribution takes real values on every real-valued test function and is therefore a real distribution under the convention of A distribution with a vanishing derivative is a polynomial (Locally essentially bounded activations with null discontinuity closure, Regular distribution from a locally integrable function).

[A1]

Countable Choice means every sequence of nonempty sets has a function choosing one element from each (The Axiom of Countable Choice ( ${A C}_{ω}$ )); it is assumed by [F3] and [F5].

Proof

1.1givenF1F2F3F7F9A1choose

Let $T_{σ}$ be the regular distribution induced by $σ$ . It is real by [F9]. The hypothesis of the Statement includes all real-valued test functions. Applying [F2] to that subclass gives $N \in N_{0}$ such that every real-test convolution supported in $[- 1, 1]$ is a polynomial of degree at most $N$ . By [F7], choose a nonnegative smooth bump supported in $(- 1, 1)$ and divide by its positive finite integral to obtain the real unit-mass test $ρ$ . Its rescalings $ρ_{ε}$ for $0 < ε \leq 1$ remain supported in $[- 1, 1]$ , so $g_{ε} = σ * ρ_{ε}$ is a polynomial of degree at most $N$ .

2.1step 1.1F2F3F6F8A1algebra

By [F3], the regular distributions induced by $g_{ε}$ converge to $T_{σ}$ on every test function. For every $ψ \in C_{c}^{\infty} (R)$ , the definition [F8] and this convergence give $⟨ D^{N + 1} T_{σ}, ψ ⟩ = (- 1)^{N + 1} ⟨ T_{σ}, ψ^{(N + 1)} ⟩ = \lim_{ε ↓ 0} (- 1)^{N + 1} \int g_{ε} (x) ψ^{(N + 1)} (x) d x = 0$ , because [F6] makes each integral zero for the polynomial $g_{ε}$ of degree at most $N$ . Thus $D^{N + 1} T_{σ} = 0$ .

3.1step 2.1F4F5A1∎

By [F4], $T_{σ}$ is the regular distribution of a polynomial $p$ . Since both $σ$ and $p$ are locally integrable, injectivity [F5] implies $σ = p$ almost everywhere. This proves the forward implication; its contrapositive is the equivalent existence assertion.

Depends on

Used by

Rough nonpolynomial activations are dense on compact sets Lemma

Dependency tree · two levels

49 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

Leshno, Lin, Pinkus, and Schocken, Multilayer Feedforward Networks With a Non-Polynomial Activation Function Can Approximate Any Function, proof of Theorem 1, Step 7 (standard reference, not scraped)
Semyon Dyatlov, Lecture Notes for 18.155, mollifier approximation and regular distributions (standard reference, not scraped)