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Quantile optimization in semidiscrete optimal transport

Yinchu Zhu, Ilya O. Ryzhov

TL;DR

This work introduces quantile optimization for semidiscrete optimal transport, showing how to characterize and compute the optimal transport plan when the objective is a cost quantile rather than its expectation. The authors develop a three-part analytical framework—feasibility certificates via a Farkas-type lemma, a convex dual linking feasibility to quantiles, and a characterization of tie-breaking rules required to realize mass splitting—and couple it with simulation-based algorithms (SAA and SA) that achieve the canonical $O(N^{-1/2})$ convergence rate. The approach yields a novel, partially randomized geometric partitioning in geographical applications, markedly different from the classical additively weighted Voronoi structure seen under mean-cost minimization. Collectively, the results deliver tractable methods with convergence guarantees for quantile-OT in the semidiscrete setting and reveal rich geometric structures with practical implications for partitioning and resource allocation.

Abstract

Optimal transport is the problem of designing a joint distribution for two random variables with fixed marginals. In virtually the entire literature on this topic, the objective is to minimize expected cost. This paper is the first to study a variant in which the goal is to minimize a quantile of the cost, rather than the mean. For the semidiscrete setting, where one distribution is continuous and the other is discrete, we derive a complete characterization of the optimal transport plan and develop simulation-based methods to efficiently compute it. One particularly novel aspect of our approach is the efficient computation of a tie-breaking rule that preserves marginal distributions. In the context of geographical partitioning problems, the optimal plan is shown to produce a novel geometric structure.

Quantile optimization in semidiscrete optimal transport

TL;DR

This work introduces quantile optimization for semidiscrete optimal transport, showing how to characterize and compute the optimal transport plan when the objective is a cost quantile rather than its expectation. The authors develop a three-part analytical framework—feasibility certificates via a Farkas-type lemma, a convex dual linking feasibility to quantiles, and a characterization of tie-breaking rules required to realize mass splitting—and couple it with simulation-based algorithms (SAA and SA) that achieve the canonical convergence rate. The approach yields a novel, partially randomized geometric partitioning in geographical applications, markedly different from the classical additively weighted Voronoi structure seen under mean-cost minimization. Collectively, the results deliver tractable methods with convergence guarantees for quantile-OT in the semidiscrete setting and reveal rich geometric structures with practical implications for partitioning and resource allocation.

Abstract

Optimal transport is the problem of designing a joint distribution for two random variables with fixed marginals. In virtually the entire literature on this topic, the objective is to minimize expected cost. This paper is the first to study a variant in which the goal is to minimize a quantile of the cost, rather than the mean. For the semidiscrete setting, where one distribution is continuous and the other is discrete, we derive a complete characterization of the optimal transport plan and develop simulation-based methods to efficiently compute it. One particularly novel aspect of our approach is the efficient computation of a tie-breaking rule that preserves marginal distributions. In the context of geographical partitioning problems, the optimal plan is shown to produce a novel geometric structure.
Paper Structure (25 sections, 19 theorems, 154 equations, 3 figures, 1 table)

This paper contains 25 sections, 19 theorems, 154 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Modeling and analysis
  3. First building block: feasibility determination
  4. Quantile optimization
  5. Characterization of tie-breaking rules
  6. Algorithms and performance guarantees
  7. Learning the optimal quantile
  8. Learning the tie-breaking rule
  9. Application: geographical partitioning
  10. Conclusion
  11. Appendix: proofs
  12. Proof of Theorem \ref{['thm:farkas']}
  13. Proof of Proposition \ref{['prop:binary']}
  14. Proof of Proposition \ref{['prop:continuity']}
  15. Proof of Proposition \ref{['prop:monotonicity']}
  16. ...and 10 more sections

Key Result

Theorem 2.1

The system described by (eq:equality) and (eq:target)-(eq:nonneg) has a solution $\pi$ if and only if there is no solution $\left(\zeta,\phi,\psi\right)$ to the system satisfying

Figures (3)

  • Figure 1: Illustrative example of optimal assignments for $c\left(x,k\right) = \|x-x_k\|_2$.
  • Figure 2: Empirical densities of $c\left(X,Y\right)$ for different values of $\alpha$. The mean-minimizing partition is included in each plot for easy visual reference.
  • Figure 3: Illustrative example of optimal assignments for $c\left(x,k\right) = \|x-x_k\|_1$.

Theorems & Definitions (19)

  • Theorem 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 9 more