LemmaStatement: Literature-sourcedProof: 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.

Signwise first-variation conditions minimize the relaxed objective

Statement

Assume AC. Let $(Θ, T)$ be a measurable space, let $H$ be a real Hilbert space, let $R : H \to [0, \infty)$ be convex and differentiable, let $φ : Θ \to H$ be bounded and the pointwise norm limit on all of $Θ$ of a sequence of $T$ -measurable finite-valued simple functions, and let $\tilde{V} : Θ \to [0, \infty)$ be bounded and measurable. For a finite signed measure $ν = ν^{+} - ν^{-}$ , put $m_{ν} = \int φ d ν$ , $G_{+} (θ) = d R_{m_{ν}} [φ (θ)] + \tilde{V} (θ)$ , and $G_{-} (θ) = - d R_{m_{ν}} [φ (θ)] + \tilde{V} (θ)$ . If $G_{+}, G_{-} \geq 0$ everywhere, $G_{+} = 0$ $ν^{+}$ -almost everywhere, and $G_{-} = 0$ $ν^{-}$ -almost everywhere, then $ν$ globally minimizes $J (λ) = R (\int φ d λ) + \int \tilde{V} d ∣ λ ∣$ over all finite signed measures $λ$ .

Statement references

Relaxed objective for a partially one-homogeneous projection

Partially one-homogeneous signed measure projection

Feature-measure energy and its regularity assumptions

The Axiom of Choice

Jordan decomposition of a signed measure into unique mutually singular positive parts

Chizat and Bach, Proposition 3.1 and Appendix C.1

Ambrosio, Gigli, and Savaré, Chapter 10

Facts & Assumptions

Given: AC, the signed measures, convex risk, feature, penalty, and signwise conditions in the Statement.

[A1]

AC supplies the Jordan decompositions of the finite signed measures $λ$ and $ν$ . The Axiom of Choice Jordan decomposition of a signed measure into unique mutually singular positive parts

[F1]

$∣ λ ∣ = λ^{+} + λ^{-}$ is the total-variation measure of a finite signed measure, and $λ = λ^{+} - λ^{-}$ . Relaxed objective for a partially one-homogeneous projection

[F2]

Bounded linear maps commute with Bochner integration, so $d R_{m_{ν}} [\int φ d (λ - ν)] = \int d R_{m_{ν}} [φ] d (λ - ν)$ . Bounded linear maps commute with Bochner integration

Proof

Given: $ν, λ$ , and the assumptions in the Statement.

1.1A1F1F2givenalgebra

The everywhere simple approximation makes $φ$ strongly measurable for every finite Jordan part, and boundedness makes each of its Bochner moments finite by the relaxed-objective convention [F1]. By [A1], write $λ = λ^{+} - λ^{-}$ and $ν = ν^{+} - ν^{-}$ . Convexity and differentiability of $R$ give $R (m_{λ}) \geq R (m_{ν}) + d R_{m_{ν}} [m_{λ} - m_{ν}]$ . By [F2], add the penalty difference and write each signed measure as its Jordan parts to obtain $J (λ) - J (ν) \geq \int G_{+} d λ^{+} + \int G_{-} d λ^{-} - \int G_{+} d ν^{+} - \int G_{-} d ν^{-} .$

2.1A1F1F2givenalgebrastep 1.1∎

The first two integrals are nonnegative because $G_{\pm} \geq 0$ and $λ^{\pm}$ are nonnegative; the last two vanish by the assumed almost-everywhere equalities. Thus $J (λ) \geq J (ν)$ for every finite signed $λ$ , proving global minimality.

Depends on

Used by

A partially homogeneous signed projection limit is a minimizer Theorem

Dependency tree · two levels

24 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)