# Direct Estimation of Information Divergence Using Nearest Neighbor Ratios

## TL;DR

This paper introduces a graph-based method for directly estimating Rényi and f-divergences from sample data, achieving optimal convergence rates and improved computational efficiency over existing techniques.

## Contribution

The authors develop a novel graph-theoretic estimator for divergence measures that attains parametric convergence rates and is more computationally efficient than previous methods.

## Key findings

- Estimator achieves MSE rate of O(N^{-2γ/(γ+d)}) for γ-Hölder smooth functions.
- Ensemble estimator attains parametric MSE rate of O(1/N) under certain conditions.
- Method is computationally more tractable than competing divergence estimators.

## Abstract

We propose a direct estimation method for R\'{e}nyi 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 $\eta:=M/N$ 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 R\'{e}nyi 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 $\gamma$-H\"{o}lder smooth functions, the estimator achieves the MSE rate of $O(N^{-2\gamma/(\gamma+d)})$. 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 estimators are more computationally tractable than other competing estimators, which makes them appealing in many practical applications.