function [K] = KPP_kNN_k_estimation(Y1,Y2,co)
%Estimates the probability product kernel of two distributions from which we have samples, Y1 and Y2. The estimation is based on k-nearest neighbors (S={k}).
%
%We use the naming convention 'K_estimation' to ease embedding new kernels on distributions.
%
%INPUT:
% Y1: Y1(:,t) is the t^th sample from the first distribution.
% Y2: Y2(:,t) is the t^th sample from the second distribution. Note: the number of samples in Y1 [=size(Y1,2)] and Y2 [=size(Y2,2)] can be different.
% co: estimator object of a kernel on distributions.
%
%REFERENCE:
% Barnabas Poczos and Liang Xiong and Dougal Sutherland and Jeff Schneider. Support Distribution Machines. Technical Report, 2012. "http://arxiv.org/abs/1202.0302" (k-nearest neighbor based estimation)
% Tony Jebara, Risi Kondor, and Andrew Howard. Probability product kernels. Journal of Machine Learning Research, 5:819-844, 2004. (probability product kernels --spec--> Bhattacharyya kernel)
%
%Copyright (C) 2012 Zoltan Szabo ("http://nipg.inf.elte.hu/szzoli", "szzoli (at) cs (dot) elte (dot) hu")
%
%This file is part of the ITE (Information Theoretical Estimators) Matlab/Octave toolbox.
%
%ITE is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by
%the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
%
%This software is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
%MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
%
%You should have received a copy of the GNU General Public License along with ITE. If not, see .
%co.mult:OK.
%verification:
if size(Y1,1)~=size(Y2,1)
error('The dimension of the samples in Y1 and Y2 must be equal.');
end
if co.p %[p(x)dx]
K = estimate_Dtemp2(Y1,Y2,co);
else %[q(x)dx]
K = estimate_Dtemp2(Y2,Y1,co);
end