function [fu_CMN, fu_1NN, fu_N_1NN] = harmonic_function(W, fl)
% [fu, fu_CMN] = harmonic_function(W, fl)
%
% Semi-supervised learning with basic harmonic function.
% That is eq (5) in "Semi-Supervised Learning Using Gaussian Fields 
% and Harmonic Functions".  Xiaojin Zhu, Zoubin Ghahramani, John Lafferty.  
% The Twentieth International Conference on Machine Learning (ICML-2003).
%
% Input:
%   W: n*n weight matrix.  The first L entries(row,col) are for labeled data,
%      the rest for unlabeled data.  W has to be symmetric, and all
%      entries has to be non-negative.  Also note the graph may be disconnected,
%      but each connected subgraph has to have at least one labeled point.
%      This is to make sure the sub-Laplacian matrix is invertible.
%   fl: L*c label matrix.  Each line is for a labeled point, in 
%      one-against-all encoding (all zero but one 1).  For example in binary
%      classification each line would be either "0 1" or "1 0".
%
% Output:
%   fu: The harmonic solution, i.e. eq (5) in the ICML paper.
%      (n-L)*c label matrix for the unlabeled points.  Each row is for
%      an unlabeled point, and each column for a class.  The class with the 
%      largest value is the predicted class of the unlabeled point.
%   fu_CMN: The solution after applying Class Mass Normalization (CMN), as
%      in eq (9) of the ICML paper.  The class proportions are the maximum
%      likelihood (frequency) estimate from labeled data fl.  The CMN heuristic
%      is known to sometimes improve classification accuracy.  As before,
%      the class with the largest value in a row is the predicted class.
%
% Note:
%   If there is a warning "Matrix is singular to working precision.", 
%   and fu is all NaN, it usually means the original graph is disconnected, 
%   and some connected components have no labeled data.  To solve the problem:
%	1. Try a different graph 
% 	2. Use a different algorithm, e.g. the Gaussian Field kernels
%
% 
% Please note there is no warranty.  In fact it is not even a software, but
% merely research code.  Feel free to modify it for your work. 
% Xiaojin Zhu, zhuxj@cs.cmu.edu
% 2004
% Modified by Xiao-Ming Wu, Jan, 2014

l = size(fl, 1); % the number of labeled points
n = size(W, 1); % total number of points

% the graph Laplacian L=D-W
L = diag(sum(W)) - W;
L = sparse(L);
% the harmonic function.
fu_1NN = - (L(l+1:n, l+1:n)) \ L(l+1:n, 1:l);
fu = fu_1NN * fl;

% compute the CMN solution
q = sum(fl)+1; % the unnormalized class proportion estimate from labeled data, with Laplace smoothing
fu_CMN = fu .* repmat(q./sum(fu), n-l, 1);

% the normalized solution proposed in "Analyzing the harmonic 
% structure in graph-based learning"(Wu et al. NIPS 2013)
fu_N_1NN = fu_1NN .* repmat(1./sum(fu_1NN), n-l, 1);



