ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Moving features can represent a target missed by a fixed feature

Example

Let $H = R^{2}$ , $y = (1, 0)$ , $R (z) = \frac{1}{2} ∥ z - y ∥^{2}$ , $Θ = [0, 2 π]$ , $φ (θ) = (\cos θ, \sin θ)$ , and $V = 0$ . The one-particle measure assigning unit mass to $(a, θ)$ represents $a φ (θ)$ and can attain zero risk at $(a, θ) = (1, 0)$ . If the feature is fixed at $θ = π / 2$ and only $a$ varies, the minimum risk is $1 / 2$ . This finite-dimensional feature-energy example shows a representational distinction only; it makes no convergence claim about either training dynamics.

Example references

Feature-measure energy and its regularity assumptions

Chizat and Bach, Sections 1.1 and 2.1

Ambrosio, Gigli, and Savaré, Chapters 8 and 11

Facts & Assumptions

Given: The finite-dimensional model and risk in the Example.

[F1]

A one-particle measure at $(a, θ)$ has feature moment $a φ (θ)$ and objective $R (a φ (θ))$ . Feature-measure energy and its regularity assumptions

Verification

Given: $R$ , $y$ , $φ$ , and [F1] as in the Example.

1.1F1givenalgebra

At $(a, θ) = (1, 0)$ , $φ (0) = (1, 0) = y$ , so [F1] gives risk $R (y) = 0$ . Since $R$ is a squared norm divided by two, no objective value is negative; hence the moving model has minimum risk zero.

2.1F1givenalgebra∎

With the feature fixed at $π / 2$ , $φ (π / 2) = (0, 1)$ and $∥ a φ (π / 2) - y ∥^{2} = a^{2} + 1$ . Therefore $R (a φ (π / 2)) = (a^{2} + 1) / 2 \geq 1 / 2$ , with equality at $a = 0$ . This gives the claimed fixed-feature minimum.

Depends on

Feature-measure energy and its regularity assumptions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Lénaïc Chizat and Francis Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport (standard reference, not scraped)
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures (standard reference, not scraped)