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

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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.

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Tent-map iterates and ReLU implementation

Statement

Let $ρ (z) = \max {0, z}$ and define $τ (x) = 2 ρ (x) - 4 ρ (x - 1 / 2)$ for $x \in [0, 1]$ . For each integer $q \geq 0$ , $τ^{q}$ is affine and maps bijectively onto $[0, 1]$ on each of the $2^{q}$ dyadic intervals $[j 2^{- q}, (j + 1) 2^{- q}]$ , with slope of magnitude $2^{q}$ . It is computed on $[0, 1]$ by a ReLU network with $2 q$ hidden units, $q$ hidden layers, and $O (q + 1)$ weights; its depth, counting input and output layers, is $q + 2$ .

Statement references

Yarotsky, Proposition 2 proof: repeated tent-map implementation

Telgarsky, Lemma 3.10 and Corollary 3.9: tent-map crossing count

Telgarsky, Chapter 5: depth and approximation context

Facts & Assumptions

Given: The ReLU network and depth convention in ReLU feedforward networks and complexity conventions, the map $ρ (z) = \max {0, z}$ , and $q \in N_{0}$ .

[F1]

A ReLU unit applies $ρ$ to an affine combination of outputs of earlier units, and skip connections are allowed (ReLU feedforward networks and complexity conventions).

Proof

1.1givenF1induction

On $[0, 1 / 2]$ , $τ (x) = 2 x$ , and on $[1 / 2, 1]$ , $τ (x) = 2 - 2 x$ . Thus $τ$ maps each half interval bijectively onto $[0, 1]$ with slope magnitude $2$ . If $τ^{q}$ has the stated property, each of its dyadic intervals is split at its midpoint by the preimage of $1 / 2$ ; composition with $τ$ makes two dyadic intervals, each mapped bijectively onto $[0, 1]$ , with slopes multiplied by $2$ . Induction proves the interval and slope claims.

2.1step 1.1F1construct∎

A layer with two ReLU units computes $u = ρ (2 v)$ and $w = ρ (2 v - 1)$ from the preceding scalar value $v \in [0, 1]$ , and the affine output $u - 2 w$ equals $τ (v)$ . For each next block substitute the preceding block's affine output expression for $v$ in its two preactivations, so one hidden layer suffices per iterate. Forward each block's affine output to the final affine unit; these skip edges add only $O (q)$ parameters. The network uses $2 q$ hidden units and $O (q + 1)$ weights; counting input and output gives depth $q + 2$ .

Depends on

ReLU feedforward networks and complexity conventions

Used by

A single high-degree gate defeats the stated small-k separation Counterexample
Tent-map composition gives exponentially many intervals Example
The unrestricted all-parameter depth separation is false False statement
Logarithmic-size ReLU approximation of the square Lemma
Depth separation with explicit semi-algebraic crossing budgets Theorem

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

Dmitry Yarotsky, Error bounds for approximations with deep ReLU networks, Proposition 2 (standard reference, not scraped)
Matus Telgarsky, Benefits of Depth in Neural Networks, Lemma 3.10 and Corollary 3.9 (standard reference, not scraped)
Matus Telgarsky, Deep learning theory lecture notes, Chapter 5 (standard reference, not scraped)