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The selected pointwise field is a strong Wasserstein subgradient

Statement

Assume AC and the feature-measure energy hypotheses. Fix $r > 0$ and $μ \in P_{2} (Ω)$ concentrated on $Q_{r}$ . Define $g_{μ} (u) = S (\nabla R (\int Φ d μ), u)$ , the minimum-norm element of $d Φ_{u}^{*} \nabla R (\int Φ d μ) + \partial_{λ_{V}} (V + ι_{Q_{r}}) (u)$ . For every $η \in P_{2} (Ω)$ concentrated on $Q_{r}$ and every coupling $γ$ of $μ$ and $η$ , $F (η) \geq F (μ) + \int g_{μ} (x) \cdot (y - x) d γ (x, y) + \frac{λ_{V} - C_{r}}{2} \int ∣ y - x ∣^{2} d γ (x, y), C_{r} = L_{r} A_{r}^{2} + N_{r} B_{r} .$ In particular, $g_{μ}$ is a strong Wasserstein subgradient of the localized functional $F_{r}$ , defined as $F$ on laws concentrated on $Q_{r}$ and $+ \infty$ otherwise, in the all-coupling sense, with zero remainder after the stated quadratic term. Here $A_{r} = \sup_{Q_{r}} ∥ d Φ ∥$ , $B_{r} = {Lip}_{Q_{r}} (d Φ)$ , and $L_{r}, N_{r}$ are respectively a Lipschitz constant and norm bound for $d R$ on the closed convex hull of $Φ (Q_{r})$ ; put $λ_{r} = λ_{V} - C_{r}$ . It is the unique minimum-norm vector field among all all-coupling strong subgradients in $L^{2} (μ; R^{d})$ .

Statement references

Feature-measure energy and its regularity assumptions

The localized lifted feature energy is semiconvex

The localized lifted subdifferential is a pointwise constrained subdifferential

Generalized W2 geodesics and strong Wasserstein subgradients

Chizat and Bach, Lemma B.2

Ambrosio, Gigli, and Savaré, Proposition 10.4.2

Facts & Assumptions

Given: AC, $r$ , $μ, η$ , and a coupling $γ$ as in the Statement.

[F1]

The localized lifted energy has semiconvexity coefficient $λ_{V} - C_{r}$ . The localized lifted feature energy is semiconvex

[F2]

The localized lifted subdifferential equals the smooth feature gradient plus the pointwise constrained potential subdifferential, and its minimum section is measurable. The localized lifted subdifferential is a pointwise constrained subdifferential

[F3]

The first marginal of a coupling is $μ$ and the second is $η$ . The Euclidean quadratic Wasserstein space

[F4]

AC supplies Countable Choice for the inherited Bochner approximation branches. Measurability and the norm bound for the full shifted minimum section are supplied by [F2]. The Axiom of Choice AC implies DC implies countable choice The Axiom of Countable Choice ( ${A C}_{ω}$ ) Bochner dominated convergence theorem

Proof

Given: AC and the data above.

1.1F2F3F4construct

On $(R^{d} \times R^{d}, γ)$ , set $X (x, y) = x$ and $Y (x, y) = y$ . This is a Euclidean Borel probability label space of dimension $2 d$ , with output dimension $d$ ; the arbitrary-label lift and subdifferential formula therefore apply. These are admissible lifts because both marginals are concentrated on $Q_{r}$ . Their laws are $μ$ and $η$ by [F3], so their lifted energies are $F (μ)$ and $F (η)$ . The pointwise minimum section and smooth gradient in [F2] give the lifted subgradient $s (x, y) = g_{μ} (x) \in L^{2} (γ; R^{d})$ .

2.1F1step 1.1

Apply the semiconvex subgradient inequality [F1] to the two lifts. The Hilbert inner product is $\int g_{μ} (x) \cdot (y - x) d γ$ and the squared displacement norm is $\int ∣ y - x ∣^{2} d γ$ . Substitution gives exactly the asserted inequality for this coupling. Since $γ$ was arbitrary, it holds for every coupling and proves the strong-subgradient claim.

3.1F1F2step 2.1∎

Conversely let $h \in L^{2} (μ)$ satisfy the all-coupling strong-subgradient inequality. Lift on $(R^{d}, μ)$ with $X (x) = x$ , and take the coupling $(X, (1 - t) X + t Y)_{#} μ$ for any admissible $Y$ . Its displacement norm is $t ∥ Y - X ∥_{2}$ . Dividing the strong inequality by $t$ gives the lower bound $⟨ h, Y - X ⟩$ for the directional derivative of $E_{r}$ at $X$ in direction $Y - X$ . Semiconvexity of $E_{r}$ implies its secant slope is at least that directional derivative plus $λ_{r} ∥ Y - X ∥_{2}^{2} / 2$ ; consequently $h \in \partial_{λ_{r}} E_{r} (X)$ . The pointwise formula [F2] gives $h (x) \in d Φ_{x}^{*} \nabla R (m_{μ}) + \partial_{λ_{V}} (V + ι_{Q_{r}}) (x)$ almost everywhere. The full shifted minimum has norm no larger pointwise, so $∥ g_{μ} ∥_{2} \leq ∥ h ∥_{2}$ . Equality forces equality pointwise by strict convexity and the projection inequality. This proves the claimed minimality and uniqueness.

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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, Gigli, and Giuseppe Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures (standard reference, not scraped)