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Optimal quantization with branched optimal transport distances

Paul Pegon, Mircea Petrache

TL;DR

This work introduces branched quantization, marrying optimal quantization with branched transport costs that induce network-like, fractal interfaces. It establishes a Gamma-convergence framework yielding a branched-transport Zador-type limit, showing that the centers of optimal $N$-point quantizers distribute according to $M_{\alpha,d}(\nu)^{-1} \nu^{\frac{\alpha}{\alpha+1/d}}$ and that the quantization error scales as $\mathcal{E}^\alpha(\nu,N) \sim c_{\alpha,d} M_{\alpha,d}(\nu)^{\alpha+1/d} N^{-\beta/d}$ with $\beta=1+d\alpha-d$. The paper also proves uniform Delone-type bounds for the atoms when $\nu$ is $d$-Ahlfors regular, and introduces a landscape function with $\beta$-Hölder regularity, yielding control over branched Voronoi basins whose boundaries are conjectured to be fractal. Together these results provide a rigorous foundation for branched quantization and reveal the macroscopic distribution and microscopic geometry of optimal mass-transport networks in this setting.

Abstract

We consider the problem of optimal approximation of a target measure by an atomic measure with $N$ atoms, in branched optimal transport distance. This is a new branched transport version of optimal quantization problems. New difficulties arise, since in classical semi-discrete optimal transport with Wasserstein distance, the interfaces between cells associated with neighboring atoms have Voronoi structure and satisfy an explicit description. This description is missing for our problem, in which the cell interfaces are thought to have fractal boundary. We study the asymptotic behaviour of optimal quantizers for absolutely continuous measures as the number $N$ of atoms grows to infinity. We compute the limit distribution of the corresponding point clouds and show in particular a branched transport version of Zador's theorem. Moreover, we establish uniformity bounds of optimal quantizers in terms of separation distance and covering radius of the atoms, when the measure is $d$-Ahlfors regular. A crucial technical tool is the uniform in $N$ Hölder regularity of the landscape function, a branched transport analog to Kantorovich potentials in classical optimal transport.

Optimal quantization with branched optimal transport distances

TL;DR

This work introduces branched quantization, marrying optimal quantization with branched transport costs that induce network-like, fractal interfaces. It establishes a Gamma-convergence framework yielding a branched-transport Zador-type limit, showing that the centers of optimal -point quantizers distribute according to and that the quantization error scales as with . The paper also proves uniform Delone-type bounds for the atoms when is -Ahlfors regular, and introduces a landscape function with -Hölder regularity, yielding control over branched Voronoi basins whose boundaries are conjectured to be fractal. Together these results provide a rigorous foundation for branched quantization and reveal the macroscopic distribution and microscopic geometry of optimal mass-transport networks in this setting.

Abstract

We consider the problem of optimal approximation of a target measure by an atomic measure with atoms, in branched optimal transport distance. This is a new branched transport version of optimal quantization problems. New difficulties arise, since in classical semi-discrete optimal transport with Wasserstein distance, the interfaces between cells associated with neighboring atoms have Voronoi structure and satisfy an explicit description. This description is missing for our problem, in which the cell interfaces are thought to have fractal boundary. We study the asymptotic behaviour of optimal quantizers for absolutely continuous measures as the number of atoms grows to infinity. We compute the limit distribution of the corresponding point clouds and show in particular a branched transport version of Zador's theorem. Moreover, we establish uniformity bounds of optimal quantizers in terms of separation distance and covering radius of the atoms, when the measure is -Ahlfors regular. A crucial technical tool is the uniform in Hölder regularity of the landscape function, a branched transport analog to Kantorovich potentials in classical optimal transport.
Paper Structure (25 sections, 21 theorems, 174 equations)

This paper contains 25 sections, 21 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Branched optimal transport and optimal quantization motivations
  3. Main results
  4. Structure of the paper
  5. Background and preliminaries
  6. Background in branched optimal transport
  7. The optimal quantization problem and mass-optimal quantizers
  8. The optimal partition problem and equivalence with optimal quantization
  9. Asymptotic energy scaling and asymptotic constant
  10. Gamma-convergence and Zador-type Theorem
  11. A Gamma-convergence result
  12. Proof of the $\Gamma-\liminf$ inequality \ref{['Gamma_liminf']}.
  13. Proof of the $\Gamma-\limsup$ inequality \ref{['Gamma_limsup']}.
  14. Asymptotics of the quantization error and support of optimal quantizers
  15. Equidistribution results at the macroscopic scale
  16. ...and 10 more sections

Key Result

Theorem 1

Let $\nu \in \mathscr{M}^+_c(\mathbb{R}^d)$ be a measure which is absolutely continuous with respect to the Lebesgue measure ${\mathscr L}^d$, and let $\alpha\in(1-1/d, 1)$.

Theorems & Definitions (45)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Simplified statement of \ref{['thm:holderz']}
  • Proposition 4: colomboStabilityMailingProblem2019
  • Definition 5: Optimal traffic plan
  • Proposition 6
  • Definition 7: Optimal quantizer
  • Theorem 8
  • proof
  • Definition 9: Mass-optimal quantizer
  • ...and 35 more