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Stationarity of a partially homogeneous signed projection limit

Statement

Assume AC, a separable real Hilbert space $H$ , and the feature-energy hypotheses: $R$ is convex and differentiable with locally Lipschitz differential bounded on sublevels, $Φ$ is Fr\acute{e}chet differentiable, $V$ is semiconvex, and the nested cutoff sets and derivative-growth bounds in the feature-energy definition hold. Let $d \geq 2$ , $Θ \subseteq R^{d - 1}$ , $φ : Θ \to H$ be bounded and continuously differentiable with Lipschitz differential, and $\tilde{V} : Θ \to [0, \infty)$ be bounded and continuously differentiable with Lipschitz differential; set $Φ (w, θ) = w φ (θ)$ and $V (w, θ) = ∣ w ∣ \tilde{V} (θ)$ . For every $f \in H$ , require dense regular values for $G_{f} (θ) = ⟨ f, φ (θ) ⟩ + \tilde{V} (θ)$ . Retain either (i) $Θ$ is the closure of a nonempty bounded connected open convex set with $C^{1}$ boundary and $d G_{f} (θ) [n_{θ}] = 0$ on $\partial Θ$ for every $f \in H$ , or (ii) $Θ = R^{d - 1}$ and for every $f \in H$ , $G_{f} (r \cdot)$ converges uniformly in $C^{1} (S^{d - 2})$ to a limit with dense regular values as $r \to \infty$ . Write $Φ (w, θ) = w φ (θ)$ , $V (w, θ) = ∣ w ∣ \tilde{V} (θ)$ , and let $μ_{t}$ be a Wasserstein gradient flow. Suppose the finite signed measures $h^{1} (μ_{t})$ converge weakly on $Θ$ to a finite signed measure $ν$ , where weak convergence means convergence against every function in $C_{b} (Θ)$ . Define $m_{ν} : = \int φ d ν$ by the signed Bochner-integral convention for the relaxed objective. Put $a_{ν} (θ) = d R_{m_{ν}} [φ (θ)]$ , $G_{+} (θ) = a_{ν} (θ) + \tilde{V} (θ)$ , and $G_{-} (θ) = - a_{ν} (θ) + \tilde{V} (θ)$ . Then $G_{+} = 0$ for $ν^{+}$ -almost every point and $G_{-} = 0$ for $ν^{-}$ -almost every point, where $ν = ν^{+} - ν^{-}$ is the Jordan decomposition. This is the sign-correct formulation of stationarity for the reduced functional $J (ν) = R (\int φ d ν) + \int \tilde{V} d ∣ ν ∣$ .

Statement references

Feature-measure energy and its regularity assumptions

Wasserstein gradient flow for a feature-measure energy

Partially one-homogeneous signed measure projection

Relaxed objective for a partially one-homogeneous projection

Feature and energy identities for homogeneous projections

Chizat and Bach, Lemma C.6 and Appendix A.2.1

Ambrosio, Gigli, and Savaré, Chapter 11

Facts & Assumptions

Given: AC and the full hypotheses in the Statement.

[F1]

Signed narrow convergence on the closed Euclidean domain implies norm convergence of bounded continuous Hilbert-valued moments, BL convergence, and positive-Jordan-part lower semicontinuity. Signed narrow convergence controls variation tails and Bochner moments

[F2]

The actual flow satisfies the all-endpoint energy and radial identities, finite total action, and $S (t) \leq S (0) + C t$ when its feature moments are bounded. Partial feature flows have exact energy and weight-moment identities

[F3]

The raw projections $α_{t}, β_{t}$ are nonnegative and $σ_{t} = h^{1} (μ_{t}) = α_{t} - β_{t}$ ; their difference's Jordan parts need not equal them. Partially one-homogeneous signed measure projection

[F4]

Cauchy–Schwarz bounds products of square-integrable functions. Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs

Proof

Given: The flow and signed projection in the Statement.

1.1F1F2given

Write $m_{t} = \int φ d σ_{t}$ and $f_{t} = \nabla R (m_{t})$ . By [F1], $m_{t} \to m_{ν}$ in norm, so $f_{t} \to f_{ν}$ in norm and $G_{\pm, t} = \pm ⟨ f_{t}, φ ⟩ + \tilde{V}$ converge uniformly to $G_{\pm}$ . The functions have a common finite magnitude bound $B \geq 1$ . The moments are uniformly bounded for all $t \geq 0$ : convergence bounds the tail, and finite-horizon continuity from [F2] bounds the remaining compact interval. Thus [F2] gives $S (t) \leq S (0) + C t$ and $\int_{0}^{\infty} A (t) d t < \infty$ , where $A (t) = \int ∣ v_{t} ∣^{2} d μ_{t}$ .

2.1F1F3step 1.1construct

Suppose $G_{+}$ is nonzero on a set of positive $ν^{+}$ measure. Regularity yields $δ > 0$ and $ψ \in C_{c} (Θ)$ , $0 \leq ψ \leq 1$ , supported in ${∣ G_{+} ∣ > 2 δ}$ , with $c_{0} = \int ψ d ν^{+} > 0$ . Indeed choose a compact subset of a positive-measure level set and a continuous distance cutoff in that open set. For late $t$ , $∣ G_{+, t} ∣ \geq δ$ on this support. Since $α_{t} \geq σ_{t}^{+}$ (use the partition formula for the positive part of $α_{t} - β_{t}$ ), [F1] gives ${lim inf}_{t \to \infty} \int ψ d α_{t} \geq c_{0}$ . This asserts no convergence of either raw component.

3.1F2F3F4step 1.1step 2.1algebra

Put $L (t) = \int_{w > 0} w G_{+, t}^{2} d μ_{t} + \int_{w < 0} (- w) G_{-, t}^{2} d μ_{t}$ . Step 2.1 implies $L (t) \geq δ^{2} c_{0} / 2$ eventually. Off $w = 0$ , the weight component of the flow velocity is $- G_{+, t}$ for positive weights and $G_{-, t}$ for negative weights. Therefore [F4] gives $L (t)^{2} \leq (\int_{w \neq 0} G_{sign (w), t}^{2} d μ_{t}) (\int_{w \neq 0} w^{2} G_{sign (w), t}^{2} d μ_{t}) \leq A (t) B^{2} S (t) .$ Thus $A (t) \geq c / (S (0) + C t)$ for almost every sufficiently late $t$ . A zero denominator contradicts $L (t) > 0$ ; a positive constant denominator already forces infinite action, and a growing linear denominator has divergent reciprocal integral. Every case contradicts [F2]. Hence $G_{+} = 0$ $ν^{+}$ -almost everywhere.

4.1F1F2F3F4step 3.1∎

Apply steps 2.1–3.1 with $- σ_{t} = β_{t} - α_{t}$ , its positive limit part $ν^{-}$ , and $G_{-}$ in place of $G_{+}$ . This proves $G_{-} = 0$ $ν^{-}$ -almost everywhere, completing both sign conclusions.

Source note

The action is integrated against $μ_{t}$ , whereas the projection carries the factor $∣ w ∣$ . The radial second-moment identity and the Cauchy–Schwarz estimate bridge those measures. The proof never identifies raw components with Jordan parts, never assumes their separate convergence, and does not import an AGS stationarity theorem.

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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)