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About this paper
Bharath K. Sriperumbudur, Kenji Fukumizu, Arthur Gretton, Bernhard Schölkopf, Gert R. G. Lanckriet
The authors develop integral probability metrics (IPMs) for comparing distributions from finite samples by taking the largest difference in expectations over a chosen function class. They relate IPMs to phi-divergences, give estimators, and connect them to classification. For bounded-kernel MMD, estimation error is order m^-1/2 plus n^-1/2, and the estimator is strongly consistent. The rate has no dimension term under these assumptions.
Key insights
- IPMs and φ-divergences are essentially distinct families of probability measures, with the total variation distance being the only non-trivial measure common to both classes.
- Empirical estimators for IPMs, particularly Maximum Mean Discrepancy (MMD), are strongly consistent, computationally efficient, and often exhibit favorable, dimension-independent convergence rates from finite samples.
- IPMs can be directly interpreted as optimal risks for specific binary classification problems, establishing connections between these metrics and classifier properties such as smoothness and margins.
Summary by alphaXiv.
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