# Correntropy

## Introduction

Correntropy is a nonlinear similarity measure between two random variables.

## History

The name Correntropy comes from correlation and entropy.

## Definition

There are several possible definitions depending on the interpretation which are not necessarily equivalent. Let X,Y be two random variables.

### Ideal view

$V(X,Y) = \operatorname{E}\left[\delta(X - Y)\right]$

where δ is the Dirac delta distribution function.

### Similarity view

$V(X,Y) = \operatorname{E}\left[\kappa(X, Y)\right]$

where κ is a non-negative definite function. It is not necessary that κ is shift invariant as in the ideal view case.

### Estimator view

$\hat{V}(X,Y) = \frac{1}{N}\sum_{k=1}^N \kappa(x(k), y(k))$

where $\{(x(k),y(k))\}_{k=1}^N$ is the set of observations.