On concentration of the empirical measure for radial transport costs

Abstract

Let $μ$ be a probability measure on $\mathbb{R}^d$ and $μ_N$ its empirical measure with sample size $N$. We prove a concentration inequality for the optimal transport cost between $μ$ and $μ_N$ for radial cost functions with polynomial local growth, that can have superpolynomial global growth. This result generalizes and improves upon estimates of Fournier and Guillin. The proof combines ideas from empirical process theory with known concentration rates for compactly supported $μ$. By partitioning $\mathbb{R}^d$ into annuli, we infer a global estimate from local estimates on the annuli and conclude that the global estimate can be expressed as a sum of the local estimate and a mean-deviation probability for which efficient bounds are known.

Figures (4)

• Figure 1: Log-log plots of functions $N\mapsto \mathbb{E}[\mathcal{T}_p(\mu, \mu_N)]$ for Geometric, Poisson, Gaussian and Weibull distributions with $p=1$. The dashed line is a theoretical upper bound $C/\sqrt{N}$ with $C=25$.
• Figure 2: Log-log plots of functions $N\mapsto \mathbb{E}[\mathcal{T}_p(\mu, \mu_N)]$ for Geometric, Poission, Gaussian and Weibull distributions with $p=1,2,3$. The dashed line is a theoretical upper bound $C/\sqrt{N}$ with $C=25$.
• Figure 3: Log-log plots of functions $N\mapsto \mathbb{E}[\mathcal{E}_{p, a}(\mu, \mu_N)]$ for Geometric, Poisson, Gaussian and Weibull distributions with $p=1$ and $a=1/2^8$. The dashed line is a theoretical upper bound $C/\sqrt{N}$ with $C=0.1$.
• Figure 4: Log-log plots of functions $N\mapsto \mathbb{E}[\mathcal{E}_{p,a}(\mu, \mu_N)]$ for Gaussian and Weibull distributions with $p=1,2,3$ and $a=1/2^8$. The dashed line is a theoretical upper bound $C/\sqrt{N}$ with $C=0.01$.

Theorems & Definitions (26)

• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Proposition 3.1
• proof
• Example 3.2: $\mathscr{D}_f$ assuming compact support
• Example 3.3: $\mathcal{T}_p$ assuming finite $q$th moment
• Example 3.4: $\mathcal{T}_p$ assuming finite exponential moment
• Example 3.5: $\mathcal{E}_{p, a}$ when $p\ge 1$
• Example 3.6: $\mathcal{E}_{p,a}$ when $p\in (0,1)$