ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

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.

Many linear regions give no training or generalization guarantee

Example

Choose a one-hidden-layer ReLU network with input dimension $2$ , hidden width $10$ , and hidden preactivation hyperplanes in general position. Its hidden activation pattern has $56$ full-dimensional regions. If every output weight and the output bias are set to zero, the represented function is identically zero; on the regression distribution with $X$ uniform on $[0, 1]^{2}$ and target $Y = 1$ , its squared prediction risk is exactly $1$ .

Example references

Serra et al., §1 and Theorem 1: region count as an expressivity proxy

Telgarsky, Chapter 5: depth and approximation context

Facts & Assumptions

Given: The one-hidden-layer network, the uniform input distribution, and the constant target in the example.

[F1]

Ten hyperplanes in general position in dimension $2$ induce $\sum_{j = 0}^{2} (\binom{10}{j})$ full-dimensional regions (Hyperplane arrangement upper count and general-position count).

[F2]

The network output is an affine unit applied to input coordinates and hidden-unit values (ReLU feedforward networks and complexity conventions).

Verification

1.1givenF1algebra

By [F1], the activation cells number $(\binom{10}{0}) + (\binom{10}{1}) + (\binom{10}{2}) = 1 + 10 + 45 = 56$ .

2.1givenF2algebra∎

Set all coefficients into the affine output unit and its bias to zero. By [F2], the prediction is then $0$ at every input. Since $Y = 1$ surely, $(0 - Y)^{2} = 1$ surely, so the expected squared risk is $1$ despite the $56$ activation cells.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Serra, Tjandraatmadja, and Ramalingam, Bounding and Counting Linear Regions of Deep Neural Networks, §1 and Theorem 1 (standard reference, not scraped)
Matus Telgarsky, Deep learning theory lecture notes, Chapter 5 (standard reference, not scraped)