Direct estimation of information divergence using nearest neighbor ratios

We propose a direct estimation method for Rényi and f-divergence measures based on a new graph theoretical interpretation. Suppose that we are given two sample sets X and Y, respectively with N and M samples, where η := M/N is a constant value. Considering the k-nearest neighbor (k-NN) graph of Y in the joint data set (X, Y), we show that the average powered ratio of the number of X points to the number of Y points among all k-NN points is proportional to Rényi divergence of X and Y densities. A similar method can also be used to estimate f-divergence measures. We derive bias and variance rates, and show that for the class of γ-Hölder smooth functions, the estimator achieves the MSE rate of O(N
^-2γ/(γ+d)
). Furthermore, by using a weighted ensemble estimation technique, for density functions with continuous and bounded derivatives of up to the order d, and some extra conditions at the support set boundary, we derive an ensemble estimator that achieves the parametric MSE rate of O(1/N). Our estimator requires no boundary correction, and remarkably, the boundary issues do not show up. Our approach is also more computationally tractable than other competing estimators, which makes them appealing in many practical applications.