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Offset-free sine ridge functions cannot uniformly approximate the square function

Statement refuted

Finite sine ridge sums without hidden-unit offsets, even when an output bias is allowed, are uniformly dense in $C ([- 1, 1], R)$ .

Statement references

Leshno et al., §5: the offset-free sine boundary

Facts & Assumptions

Given: The restricted family $S (x) = c + \sum_{j = 1}^{m} a_{j} \sin (w_{j} x)$ , with $m \in N_{0}$ and real parameters.

[F1]

The full single-hidden-layer convention includes arbitrary hidden offsets and an output bias (Single-hidden-layer network realization); this counterexample removes the hidden offsets.

Counterexample

1.1givenF1algebra

Since sine is odd, $S (0) = c$ and $(S (1) + S (- 1)) / 2 = c$ .

2.1step 1.1algebra

For the even target $f (x) = x^{2}$ , uniform error at most $ε$ implies both $∣ c ∣ = ∣ S (0) - f (0) ∣ \leq ε$ and $∣ c - 1 ∣ = ∣ (S (1) + S (- 1)) / 2 - 1 ∣ \leq ε$ .

3.1step 2.1algebra∎

Every real $c$ satisfies $\max (∣ c ∣, ∣ c - 1 ∣) \geq 1 / 2$ . Thus no such sum approximates $x^{2}$ with error less than $1 / 2$ , disproving uniform density.

Depends on

Single-hidden-layer network realization

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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, §5 (standard reference, not scraped)