Function test_empirical_kernel_map

examples/empirical_kernel_map_ex.cpp:167–309

// ----------------------------------------------------------------------------------------

void test_empirical_kernel_map (
    const std::vector<sample_type>& samples,
    const std::vector<double>& labels,
    const empirical_kernel_map<kernel_type>& ekm
)
{

    std::vector<sample_type> projected_samples;

    // The first thing we do is compute the nonlinearly projected vectors using the 
    // empirical_kernel_map.  
    for (unsigned long i = 0; i < samples.size(); ++i)
    {
        projected_samples.push_back(ekm.project(samples[i]));
    }

    // Note that a kernel matrix is just a matrix M such that M(i,j) == kernel(samples[i],samples[j]).
    // So below we are computing the normal kernel matrix as given by the radial_basis_kernel and the
    // input samples.  We also compute the kernel matrix for all the projected_samples as given by the 
    // linear_kernel.  Note that the linear_kernel just computes normal dot products.  So what we want to
    // see is that the dot products between all the projected_samples samples are the same as the outputs
    // of the kernel function for their respective untransformed input samples.  If they match then
    // we know that the empirical_kernel_map is working properly.
    const matrix<double> normal_kernel_matrix = kernel_matrix(ekm.get_kernel(), samples);
    const matrix<double> new_kernel_matrix = kernel_matrix(linear_kernel<sample_type>(), projected_samples);

    cout << "Max kernel matrix error: " << max(abs(normal_kernel_matrix - new_kernel_matrix)) << endl;
    cout << "Mean kernel matrix error: " << mean(abs(normal_kernel_matrix - new_kernel_matrix)) << endl;
    /*
        Example outputs from these cout statements.
        For the case where we use all samples as basis samples:
            Max kernel matrix error: 7.32747e-15
            Mean kernel matrix error: 7.47789e-16

        For the case where we use only 26 samples as basis samples:
            Max kernel matrix error: 0.000953573
            Mean kernel matrix error: 2.26008e-05


        Note that if we use enough basis samples we can perfectly span the space of input samples.
        In that case we get errors that are essentially just rounding noise (Moreover, using all the 
        samples is always enough since they are always within their own span).  Once we start 
        to use fewer basis samples we may begin to get approximation error.  In the second case we 
        used 26 and we can see that the data doesn't really lay exactly in a 26 dimensional subspace.  
        But it is pretty close.  
    */



    // Now let's do something more interesting.  The following loop finds the centroids
    // of the two classes of data.
    sample_type class1_center; 
    sample_type class2_center; 
    for (unsigned long i = 0; i < projected_samples.size(); ++i)
    {
        if (labels[i] == 1)
            class1_center += projected_samples[i];
        else