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Optimal Matching Problem on the Boolean Cube

Shi Feng

TL;DR

This work characterizes the expected Wasserstein-1 distance between the empirical distribution on the Boolean cube $\mathbb{B}=\oldsymbol{-1,1\boldsymbol}$ and the uniform measure, across multiple regimes of the sample size $N$. Combining Fourier analysis on the Boolean cube with large-deviation techniques, it derives precise upper and lower bounds for $\mathbb{E}[\mathsf{W}_1(\mu_N,\mu)]$ in regimes ranging from small to very large $N$ (including $N$ proportional to $2^n$ and $N=e^{\lambda n}$). The paper also establishes general properties of the matching problem, including a best-guess measure, monotonicity in $N$, and concentration/variance bounds, and demonstrates that the uniform measure often minimizes expected matching distance under Haar-invariant group actions. These results provide a detailed, regime-specific understanding of optimal matching on the Boolean cube and illuminate contrasts with continuous spaces, with potential implications for high-dimensional discrete sampling and related combinatorial optimization problems.

Abstract

We establish upper and lower bounds for the expected Wasserstein distance between the random empirical measure and the uniform measure on the Boolean cube. Our analysis leverages techniques from Fourier analysis, following the framework introduced in \cite{bobkov2021simple}, as well as methods from large deviations theory.

Optimal Matching Problem on the Boolean Cube

TL;DR

This work characterizes the expected Wasserstein-1 distance between the empirical distribution on the Boolean cube and the uniform measure, across multiple regimes of the sample size . Combining Fourier analysis on the Boolean cube with large-deviation techniques, it derives precise upper and lower bounds for in regimes ranging from small to very large (including proportional to and ). The paper also establishes general properties of the matching problem, including a best-guess measure, monotonicity in , and concentration/variance bounds, and demonstrates that the uniform measure often minimizes expected matching distance under Haar-invariant group actions. These results provide a detailed, regime-specific understanding of optimal matching on the Boolean cube and illuminate contrasts with continuous spaces, with potential implications for high-dimensional discrete sampling and related combinatorial optimization problems.

Abstract

We establish upper and lower bounds for the expected Wasserstein distance between the random empirical measure and the uniform measure on the Boolean cube. Our analysis leverages techniques from Fourier analysis, following the framework introduced in \cite{bobkov2021simple}, as well as methods from large deviations theory.

Paper Structure

This paper contains 14 sections, 11 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Wasserstein distance
  4. Fourier transform
  5. Connection between 2.1 and 2.2
  6. Total variation distance
  7. For (very) large size $N$
  8. For small size $N$
  9. For intermediate size $N$
  10. General properties of the matching problem
  11. A best guess for the empirical distribution
  12. Monotonicity of the expected matching distance
  13. Variance and concentration inequalities
  14. Conclusion

Key Result

Theorem 1.1

When $N = n^{\alpha}$ for $\alpha\geq 2$, we have When $N = e^{\lambda n}$ for $0<\lambda<\log(2)$, define $H(x) = -x\log(x)-(1-x)\log(1-x)$ and take $r^* = H^{-1}(\log(2)-\lambda) \in (0,\frac{1}{2})$. We have When $N = c2^n$, we have When $N = c(n)2^n$ for some $c(n) \to \infty$, we have

Theorems & Definitions (22)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • proof : proof of (4) and (5)
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • proof : proof of (2)
  • Lemma 5.1
  • ...and 12 more