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A W2 limit in the partial one-homogeneous case is globally minimizing

Statement

Assume AC, the full partially one-homogeneous hypotheses, one of the two source boundary alternatives, and the separated bounded-box initialization in the preceding projected-limit theorem. If the Wasserstein gradient flow $μ_{t}$ converges in $W_{2}$ to $μ_{\infty} \in P_{2} (R \times Θ)$ , then $μ_{\infty}$ is a global minimizer of $F$ over finite nonnegative measures.

Statement references

A partially homogeneous signed projection limit is a minimizer

W2 convergence implies weak convergence of homogeneous projections

Partial one-homogeneous energy is continuous under W2 convergence

Chizat and Bach, Theorem 3.5

Facts & Assumptions

Given: AC, the assumptions in the Statement, and W2 convergence to $μ_{\infty}$ .

[A1]

Assume the Axiom of Choice, as stated in the hypotheses and required by the flow and coupling suppliers. The Axiom of Choice

[F1]

W2 convergence implies weak convergence of the signed $h^{1}$ projections. W2 convergence implies weak convergence of homogeneous projections

[F2]

The partial projected-limit theorem identifies the limiting signed projection as a minimizer of $J$ and gives convergence of the flow energies to $\inf J = \inf F$ . A partially homogeneous signed projection limit is a minimizer Relaxed objective for a partially one-homogeneous projection

[F3]

The partial feature energy is continuous under W2 convergence. Partial one-homogeneous energy is continuous under W2 convergence

Proof

Given: AC and the data in the Statement.

1.1A1F1F2given

By [F1], $h^{1} (μ_{t})$ converges weakly to $h^{1} (μ_{\infty})$ . Apply [F2] to obtain $F (μ_{t}) \to \inf_{λ \geq 0} F (λ)$ .

2.1A1F2F3step 1.1∎

By [F3], $F (μ_{t}) \to F (μ_{\infty})$ along the same W2-convergent flow. Therefore $F (μ_{\infty}) = \inf_{λ \geq 0} F (λ)$ , which is the asserted global minimality.

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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)