Electronic Journal of Statistics

On the empirical estimation of integral probability metrics

Bharath K. Sriperumbudur, Kenji Fukumizu, Arthur Gretton, Bernhard Schölkopf, and Gert R. G. Lanckriet

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Given two probability measures, $\mathbb{P}$ and $\mathbb{Q}$ defined on a measurable space, $S$, the integral probability metric (IPM) is defined as $$\gamma_{\mathcal{F}}(\mathbb{P},\mathbb{Q})=\sup\left\{\left\vert \int_{S}f\,d\mathbb{P}-\int_{S}f\,d\mathbb{Q}\right\vert\,:\,f\in\mathcal{F}\right\},$$ where $\mathcal{F}$ is a class of real-valued bounded measurable functions on $S$. By appropriately choosing $\mathcal{F}$, various popular distances between $\mathbb{P}$ and $\mathbb{Q}$, including the Kantorovich metric, Fortet-Mourier metric, dual-bounded Lipschitz distance (also called the Dudley metric), total variation distance, and kernel distance, can be obtained.

In this paper, we consider the problem of estimating $\gamma_{\mathcal{F}}$ from finite random samples drawn i.i.d. from $\mathbb{P}$ and $\mathbb{Q}$. Although the above mentioned distances cannot be computed in closed form for every $\mathbb{P}$ and $\mathbb{Q}$, we show their empirical estimators to be easily computable, and strongly consistent (except for the total-variation distance). We further analyze their rates of convergence. Based on these results, we discuss the advantages of certain choices of $\mathcal{F}$ (and therefore the corresponding IPMs) over others—in particular, the kernel distance is shown to have three favorable properties compared with the other mentioned distances: it is computationally cheaper, the empirical estimate converges at a faster rate to the population value, and the rate of convergence is independent of the dimension $d$ of the space (for $S=\mathbb{R}^{d}$). We also provide a novel interpretation of IPMs and their empirical estimators by relating them to the problem of binary classification: while the IPM between class-conditional distributions is the negative of the optimal risk associated with a binary classifier, the smoothness of an appropriate binary classifier (e.g., support vector machine, Lipschitz classifier, etc.) is inversely related to the empirical estimator of the IPM between these class-conditional distributions.

Article information

Electron. J. Statist., Volume 6 (2012), 1550-1599.

First available in Project Euclid: 18 September 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation

Integral probability metrics empirical estimation Kantorovich metric dual-bounded Lipschitz distance (Dudley metric) kernel distance reproducing kernel Hilbert space Rademacher average Lipschitz classifier support vector machine


Sriperumbudur, Bharath K.; Fukumizu, Kenji; Gretton, Arthur; Schölkopf, Bernhard; Lanckriet, Gert R. G. On the empirical estimation of integral probability metrics. Electron. J. Statist. 6 (2012), 1550--1599. doi:10.1214/12-EJS722. https://projecteuclid.org/euclid.ejs/1347974672

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