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The localized flow cannot escape all cutoffs in finite time

Statement

Assume AC and the feature-measure energy hypotheses. Let $r_{0} > 0$ and let $μ_{0} \in P_{2} (Ω)$ have finite energy and be concentrated on $Q_{r_{0}}$ . Set $M_{0} = \sup {∥ \nabla R (z) ∥ : R (z) \leq F (μ_{0})}$ , $a = (1 + M_{0}) C_{1}$ , and $b = (1 + M_{0}) C_{2}$ . For $T < \infty$ , define $r (T) = {\begin{cases} (r_{0} + a T), & b = 0, \\ (r_{0} + a / b) e^{b T} - a / b, & b > 0. \end{cases}$ For every $R > r (T)$ , the localized flow with indicator $ι_{Q_{R}}$ remains in $Q_{r (T)}$ for $0 \leq t \leq T$ and therefore solves the continuity equation with the constraint $Ω$ alone on this interval. Localized flows with different cutoffs larger than $r (T)$ agree on $[0, T]$ . In particular no trajectory starting in $Q_{r_{0}}$ can escape every cutoff in finite time.

Statement references

Feature-measure energy and its regularity assumptions

The localized Wasserstein gradient flow exists, satisfies EVI, and is unique

The Axiom of Choice

Chizat and Bach, Proposition 2.5 and Appendix B.2.1

Ambrosio, Gigli, and Savaré, Chapter 4

Facts & Assumptions

Given: AC, the model assumptions, initial law and cutoff family.

[F1]

Along a localized flow the energy decreases, and $R (m_{μ_{t}}) \leq F (μ_{0})$ because $V \geq 0$ . The localized Wasserstein gradient flow exists, satisfies EVI, and is unique Feature-measure energy and its regularity assumptions

[F2]

The constrained minimum-norm field is no larger than the admissible field obtained by adding zero from the cutoff normal cone; on $Q_{s}$ its norm is at most $(1 + M_{0}) (C_{1} + C_{2} s)$ . Feature-measure energy and its regularity assumptions The localized Wasserstein gradient flow exists, satisfies EVI, and is unique

[F3]

If a point of $Ω$ is within distance $s$ of $Q_{r}$ , then it lies in $Q_{r + s}$ . Feature-measure energy and its regularity assumptions

[F4]

AC supplies the locally absolutely continuous label paths representing the localized flow. The Axiom of Choice The localized Wasserstein gradient flow exists, satisfies EVI, and is unique

[F5]

Solutions using the same $Ω$ -constrained velocity are unique by their W2 stability estimate. One-sided semiconvex characteristics give W2 uniqueness and stability

Proof

Given: AC, $T$ , and $μ_{0}$ as in the Statement.

1.1F1F2F4

By [F1], $m_{μ_{t}}$ stays in the $R$ -sublevel set used to define $M_{0}$ . The unconstrained part of the negative velocity has norm at most $M_{0} (C_{1} + C_{2} R) + (C_{1} + C_{2} R) = (a + b R)$ . Since $0$ belongs to the normal cone of $Q_{R}$ at each point of $Q_{R}$ , adding the cutoff constraint and taking the minimum norm cannot increase this bound by [F2]. Thus every localized label path has speed at most $a + b R$ before leaving $Q_{R}$ .

2.1F2F3step 1.1

Fix $R > r (T)$ and a label path starting at $u_{0} \in Q_{r_{0}}$ . Let $s_{R} (t) = \int_{0}^{t} ∣ {\dot{X}}_{q} ∣ d q$ . By [F3], its displacement from $u_{0}$ gives $dist (X_{t}, Q_{r_{0}}) \leq s_{R} (t)$ and hence $X_{t} \in Q_{r_{0} + s_{R} (t)}$ . At almost every time apply [F2] at its current radius (the bounded potential supporting vector at the current point remains admissible after restriction to $Q_{R}$ ), obtaining $s_{R}^{'} (t) \leq a + b (r_{0} + s_{R} (t))$ almost everywhere. If $b > 0$ , multiply by $e^{- b t}$ and integrate to get $r_{0} + s_{R} (t) \leq (r_{0} + a / b) e^{b t} - a / b$ ; if $b = 0$ , integration gives $r_{0} + s_{R} (t) \leq r_{0} + a t$ . Thus $r_{0} + s_{R} (t) \leq r (t)$ . This estimate holds for the entire localized path; in particular $r (T) < R$ places it in the relative $Ω$ -interior of $Q_{R}$ on this interval. The bound holds for almost every label, so the measure flow stays in $Q_{r (T)}$ .

3.1F3F5step 2.1∎

For $x \in Q_{r (T)}$ , the relative-neighborhood condition places an $Ω$ -neighborhood of $x$ inside $Q_{R}$ because $R > r (T)$ . Thus the cutoff and $Ω$ constraints have the same local normal cone along these paths, so each localized flow solves the equation with the common $Ω$ -constrained field. If two cutoffs exceed $r (T)$ , [F5] gives equality for the common initial law. Taking $T$ arbitrary and increasing the cutoff constructs a compatible global flow, so finite-time escape from all cutoffs is impossible.

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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)