Convergence Rates for Empirical Estimation of Binary Classification Bounds
This work provides theoretical guarantees for estimating classification bounds, which is incremental for applications in machine learning, signal processing, and information theory.
The paper tackles the problem of empirically estimating the Henze-Penrose divergence, used for bounding binary classification error rates, by deriving a convergence rate bound and concentration inequality for the Friedman-Rafsky estimator, validated experimentally on real datasets.
Bounding the best achievable error probability for binary classification problems is relevant to many applications including machine learning, signal processing, and information theory. Many bounds on the Bayes binary classification error rate depend on information divergences between the pair of class distributions. Recently, the Henze-Penrose (HP) divergence has been proposed for bounding classification error probability. We consider the problem of empirically estimating the HP-divergence from random samples. We derive a bound on the convergence rate for the Friedman-Rafsky (FR) estimator of the HP-divergence, which is related to a multivariate runs statistic for testing between two distributions. The FR estimator is derived from a multicolored Euclidean minimal spanning tree (MST) that spans the merged samples. We obtain a concentration inequality for the Friedman-Rafsky estimator of the Henze-Penrose divergence. We validate our results experimentally and illustrate their application to real datasets.