function b = dualMax2SparseAlignment(K, Kj, Y, Yj)
%A function to compute the maximum 2 sparse alignment vector between matrix
%K and vector y

%We need to go through all combinations of 2 examples, so this might be
%quite slow. 

numExamples = size(K, 1); 
sparsity = 2; 
maxAlignment = -1; 
alpha = 0.0001;

b = zeros(numExamples, 1); 
Ky = Kj'*Y;
I = alpha*eye(sparsity, sparsity); 

KySub = zeros(sparsity, 1); 
KSub = zeros(sparsity, sparsity); 
KjKjSub = zeros(sparsity, sparsity);

for i=1:numExamples
    for j=i+1:numExamples 
        
        KySub = Ky([i, j]); 
        ySub = Y([i, j]); 
        
        KjSub = Kj([i, j], [i, j]); 
        KjKjSub(1, 1) = Kj(:, i)'*Kj(:, i); 
        KjKjSub(1, 2) = Kj(:, i)'*Kj(:, j);
        KjKjSub(2, 1) = KjKjSub(1, 2); 
        KjKjSub(2, 2) = Kj(:, j)'*Kj(:, j);
        
        KjKjSub = KjKjSub + I; 
        
        bSub = KjKjSub\KySub;
        bSub = bSub/sqrt(KySub'*bSub);
        
        %Note: not really the alignment - but proportional to it 
        alignment = bSub'*KySub; 
        
        if alignment > maxAlignment 
            b = zeros(numExamples, 1); 
            b([i, j]) = bSub; 
            maxAlignment = alignment; 
        end  
    end 
end