DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Single-hidden-layer network realization

Definition

Let $d \geq 1$ and let $σ : R \to R$ be an activation function. For a finite width $m \in N_{0}$ , output weights $a_{1}, \dots, a_{m} \in R$ , input weights $w_{1}, \dots, w_{m} \in R^{d}$ , hidden biases $b_{1}, \dots, b_{m} \in R$ , and output bias $c \in R$ , the scalar single-hidden-layer realization is

$N (x) = c + \sum_{j = 1}^{m} a_{j} σ (w_{j} \cdot x + b_{j}), x \in R^{d} .$

The family $N_{σ}^{(d)}$ consists of all such functions as $m$ and the real parameters vary; $m = 0$ is allowed and gives constant functions. This is the scalar-output, one-hidden-layer specialization of a layered multilayer perceptron with linear output.

Definition references

Finite feedforward computation graphs and layered MLPs

Leshno et al., §2: network realization formula and parameter conventions

Hornik, §2: one hidden layer with linear output

Depends on

Finite feedforward computation graphs and layered MLPs

Used by

Essential boundedness does not imply pointwise uniform universality Counterexample
Offset-free sine ridge functions cannot uniformly approximate the square function Counterexample
The zero-measure Lp boundary Counterexample
One-hidden-layer particle-measure model Definition
Signed-square ReLU feature parameterization Definition
A polynomial activation confines univariate outputs to bounded degree Example
Networks approximate smooth convolutions of the activation Lemma
Polynomial activations have a bounded-degree obstruction Lemma
Rough nonpolynomial activations are dense on compact sets Lemma
Smooth nonpolynomial activations are dense on compact sets Lemma
Single-hidden-layer universal approximation Theorem

Dependency tree · one level

1 result within one dependency step 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, §2 (standard reference, not scraped)
Hornik, Approximation Capabilities of Multilayer Feedforward Networks, §2 (standard reference, not scraped)