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Absense of loops for the Wasserstein-$\mathcal{H}^1$ problem: the localization/blow-up argument

João Miguel Machado

TL;DR

The paper analyzes the variational problem for approximating a measure by a 1D network endowed with a length penalty, proving that minimizers are loop-free trees in two regimes: a finite Dirac mass mixture and an absolutely continuous bounded density. It develops a localization-blow-up framework that reduces the global problem to a limit problem on the approximate tangent space via projection onto the tangent, and uses Gamma-convergence to transfer optimality from the localized problem back to the original one. A key technical contribution is showing that loops must arise from projections and then constructing a better competitor in the limit, which yields a contradiction and eliminates loops. The results advance understanding of the topology of optimal networks for Wasserstein-based shape optimization and suggest a robust method to rule out cyclic structures in related variational problems.

Abstract

In the present work we prove that minimizers of the Wasserstein-$\mathscr{H}^1$ problem, introduced recently by Chambolle et. al., are trees in two cases: when the target measure is a sum of finitely many Dirac masses or when it has a bounded density.

Absense of loops for the Wasserstein-$\mathcal{H}^1$ problem: the localization/blow-up argument

TL;DR

The paper analyzes the variational problem for approximating a measure by a 1D network endowed with a length penalty, proving that minimizers are loop-free trees in two regimes: a finite Dirac mass mixture and an absolutely continuous bounded density. It develops a localization-blow-up framework that reduces the global problem to a limit problem on the approximate tangent space via projection onto the tangent, and uses Gamma-convergence to transfer optimality from the localized problem back to the original one. A key technical contribution is showing that loops must arise from projections and then constructing a better competitor in the limit, which yields a contradiction and eliminates loops. The results advance understanding of the topology of optimal networks for Wasserstein-based shape optimization and suggest a robust method to rule out cyclic structures in related variational problems.

Abstract

In the present work we prove that minimizers of the Wasserstein- problem, introduced recently by Chambolle et. al., are trees in two cases: when the target measure is a sum of finitely many Dirac masses or when it has a bounded density.
Paper Structure (13 sections, 10 theorems, 133 equations, 3 figures)

This paper contains 13 sections, 10 theorems, 133 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions and the localization/blow-up argument
  3. Structure of this manuscript
  4. Preliminaries
  5. Convergence of sets and measures
  6. Narrow convergence of probability measures and the Wasserstein distances
  7. Gołąb's Theorem, the length functional, blow-ups and approximate tangent spaces
  8. Loops and tree structure
  9. Absense of loops
  10. Loops are formed though projections
  11. Localizations and blow-up
  12. Better competitor and absence of loops
  13. Appendix: technical proofs of the localization/blow-up argument

Key Result

Theorem 2.2

Let ${(\Sigma_n)}_{n \in \mathbb{N}}$ be a sequence of closed and connected subsets of $\mathbb{R}^d$ converging in the sense of Kuratowski to some closed set $\Sigma$ and having locally uniform finite length, i.e. for all $R > 0$ Define the measures $\mu_n \stackrel{\hbox{\upshape\tiny def.}}{=} \mathscr{H}^1\mathbin{} \Sigma_n$, and let $\mu$ be a weak-$\star$ cluster point of this sequence. Th

Figures (3)

  • Figure 1: Heuristic proof of existence of an optimal shape for problem \ref{['problem.shape_optimization']}. If a solution has an excess part, represented in the figure by a measure having a density along $\Sigma$ and a Dirac mass, it must be formed through projections onto $\Sigma$. But then it is better to send the excess mass that is being projected to small segments in the direction of the projection.
  • Figure 2: Argument for absence of loops for \ref{['problem.shape_optimization_relaxed']}. As in the proof of existence, we begin by showing that loops are formed through projections and later use this information to construct a better competitor.
  • Figure 3: Construction of a better competitor in Thm. \ref{['theorem.no_loops_integrable_regime']}. On the right, the partition of the space into sections. For sections $i,i'$ such that $\bar{\theta}_i,\bar{\theta}_{i'}>0$ we add a segment in their direction. For $\bar{\theta}_j,\bar{\theta}_{j'}=0$ we construct a Dirac mass. On the cases of positive density we have a gain of order $\varepsilon^2$ in transportation cost, for zero density we lose $o(\varepsilon^2)$. On the left the transportation strategy of each section of the partitioned space.

Theorems & Definitions (19)

  • Definition 2.1
  • Theorem 2.2: Density version of Gołąb's Theorem
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 9 more