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356 lines
15 KiB
356 lines
15 KiB
// The contents of this file are in the public domain. See LICENSE_FOR_EXAMPLE_PROGRAMS.txt
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/*
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This is an example illustrating the use of the empirical_kernel_map
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from the dlib C++ Library.
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This example program assumes you are familiar with some general elements of
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the library. In particular, you should have at least read the svm_ex.cpp
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and matrix_ex.cpp examples.
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Most of the machine learning algorithms in dlib are some flavor of "kernel machine".
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This means they are all simple linear algorithms that have been formulated such
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that the only way they look at the data given by a user is via dot products between
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the data samples. These algorithms are made more useful via the application of the
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so-called kernel trick. This trick is to replace the dot product with a user
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supplied function which takes two samples and returns a real number. This function
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is the kernel that is required by so many algorithms. The most basic kernel is the
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linear_kernel which is simply a normal dot product. More interesting, however,
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are kernels which first apply some nonlinear transformation to the user's data samples
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and then compute a dot product. In this way, a simple algorithm that finds a linear
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plane to separate data (e.g. the SVM algorithm) can be made to solve complex
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nonlinear learning problems.
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An important element of the kernel trick is that these kernel functions perform
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the nonlinear transformation implicitly. That is, if you look at the implementations
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of these kernel functions you won't see code that transforms two input vectors in
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some way and then computes their dot products. Instead you will see a simple function
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that takes two input vectors and just computes a single real number via some simple
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process. You can basically think of this as an optimization. Imagine that originally
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we wrote out the entire procedure to perform the nonlinear transformation and then
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compute the dot product but then noticed we could cancel a few terms here and there
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and simplify the whole thing down into a more compact and easily evaluated form.
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The result is a nice function that computes what we want but we no longer get to see
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what those nonlinearly transformed input vectors are.
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The empirical_kernel_map is a tool that undoes this. It allows you to obtain these
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nonlinearly transformed vectors. It does this by taking a set of data samples from
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the user (referred to as basis samples), applying the nonlinear transformation to all
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of them, and then constructing a set of orthonormal basis vectors which spans the space
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occupied by those transformed input samples. Then if we wish to obtain the nonlinear
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version of any data sample we can simply project it onto this orthonormal basis and
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we obtain a regular vector of real numbers which represents the nonlinearly transformed
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version of the data sample. The empirical_kernel_map has been formulated to use only
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dot products between data samples so it is capable of performing this service for any
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user supplied kernel function.
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The empirical_kernel_map is useful because it is often difficult to formulate an
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algorithm in a way that uses only dot products. So the empirical_kernel_map lets
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us easily kernelize any algorithm we like by using this object during a preprocessing
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step. However, it should be noted that the algorithm is only practical when used
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with at most a few thousand basis samples. Fortunately, most datasets live in
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subspaces that are relatively low dimensional. So for these datasets, using the
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empirical_kernel_map is practical assuming an appropriate set of basis samples can be
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selected by the user. To help with this dlib supplies the linearly_independent_subset_finder.
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I also often find that just picking a random subset of the data as a basis works well.
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In what follows, we walk through the process of creating an empirical_kernel_map,
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projecting data to obtain the nonlinearly transformed vectors, and then doing a
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few interesting things with the data.
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*/
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#include <dlib/svm.h>
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#include <dlib/rand.h>
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#include <iostream>
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#include <vector>
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using namespace std;
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using namespace dlib;
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// ----------------------------------------------------------------------------------------
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// First let's make a typedef for the kind of samples we will be using.
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typedef matrix<double, 0, 1> sample_type;
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// We will be using the radial_basis_kernel in this example program.
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typedef radial_basis_kernel<sample_type> kernel_type;
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// ----------------------------------------------------------------------------------------
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void generate_concentric_circles (
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std::vector<sample_type>& samples,
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std::vector<double>& labels,
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const int num_points
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);
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/*!
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requires
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- num_points > 0
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ensures
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- generates two circles centered at the point (0,0), one of radius 1 and
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the other of radius 5. These points are stored into samples. labels will
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tell you if a given samples is from the smaller circle (its label will be 1)
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or from the larger circle (its label will be 2).
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- each circle will be made up of num_points
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!*/
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// ----------------------------------------------------------------------------------------
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void test_empirical_kernel_map (
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const std::vector<sample_type>& samples,
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const std::vector<double>& labels,
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const empirical_kernel_map<kernel_type>& ekm
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);
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/*!
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This function computes various interesting things with the empirical_kernel_map.
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See its implementation below for details.
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!*/
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// ----------------------------------------------------------------------------------------
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int main()
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{
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std::vector<sample_type> samples;
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std::vector<double> labels;
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// Declare an instance of the kernel we will be using.
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const kernel_type kern(0.1);
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// create a dataset with two concentric circles. There will be 100 points on each circle.
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generate_concentric_circles(samples, labels, 100);
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empirical_kernel_map<kernel_type> ekm;
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// Here we create an empirical_kernel_map using all of our data samples as basis samples.
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cout << "\n\nBuilding an empirical_kernel_map with " << samples.size() << " basis samples." << endl;
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ekm.load(kern, samples);
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cout << "Test the empirical_kernel_map when loaded with every sample." << endl;
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test_empirical_kernel_map(samples, labels, ekm);
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// create a new dataset with two concentric circles. There will be 1000 points on each circle.
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generate_concentric_circles(samples, labels, 1000);
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// Rather than use all 2000 samples as basis samples we are going to use the
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// linearly_independent_subset_finder to pick out a good basis set. The idea behind this
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// object is to try and find the 40 or so samples that best spans the subspace containing all the
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// data.
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linearly_independent_subset_finder<kernel_type> lisf(kern, 40);
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// populate lisf with samples. We have configured it to allow at most 40 samples but this function
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// may determine that fewer samples are necessary to form a good basis. In this example program
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// it will select only 26.
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fill_lisf(lisf, samples);
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// Now reload the empirical_kernel_map but this time using only our small basis
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// selected using the linearly_independent_subset_finder.
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cout << "\n\nBuilding an empirical_kernel_map with " << lisf.size() << " basis samples." << endl;
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ekm.load(lisf);
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cout << "Test the empirical_kernel_map when loaded with samples from the lisf object." << endl;
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test_empirical_kernel_map(samples, labels, ekm);
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cout << endl;
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}
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// ----------------------------------------------------------------------------------------
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void test_empirical_kernel_map (
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const std::vector<sample_type>& samples,
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const std::vector<double>& labels,
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const empirical_kernel_map<kernel_type>& ekm
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)
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{
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std::vector<sample_type> projected_samples;
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// The first thing we do is compute the nonlinearly projected vectors using the
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// empirical_kernel_map.
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for (unsigned long i = 0; i < samples.size(); ++i)
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{
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projected_samples.push_back(ekm.project(samples[i]));
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}
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// Note that a kernel matrix is just a matrix M such that M(i,j) == kernel(samples[i],samples[j]).
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// So below we are computing the normal kernel matrix as given by the radial_basis_kernel and the
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// input samples. We also compute the kernel matrix for all the projected_samples as given by the
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// linear_kernel. Note that the linear_kernel just computes normal dot products. So what we want to
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// see is that the dot products between all the projected_samples samples are the same as the outputs
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// of the kernel function for their respective untransformed input samples. If they match then
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// we know that the empirical_kernel_map is working properly.
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const matrix<double> normal_kernel_matrix = kernel_matrix(ekm.get_kernel(), samples);
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const matrix<double> new_kernel_matrix = kernel_matrix(linear_kernel<sample_type>(), projected_samples);
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cout << "Max kernel matrix error: " << max(abs(normal_kernel_matrix - new_kernel_matrix)) << endl;
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cout << "Mean kernel matrix error: " << mean(abs(normal_kernel_matrix - new_kernel_matrix)) << endl;
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/*
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Example outputs from these cout statements.
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For the case where we use all samples as basis samples:
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Max kernel matrix error: 7.32747e-15
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Mean kernel matrix error: 7.47789e-16
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For the case where we use only 26 samples as basis samples:
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Max kernel matrix error: 0.000953573
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Mean kernel matrix error: 2.26008e-05
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Note that if we use enough basis samples we can perfectly span the space of input samples.
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In that case we get errors that are essentially just rounding noise (Moreover, using all the
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samples is always enough since they are always within their own span). Once we start
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to use fewer basis samples we may begin to get approximation error. In the second case we
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used 26 and we can see that the data doesn't really lay exactly in a 26 dimensional subspace.
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But it is pretty close.
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*/
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// Now let's do something more interesting. The following loop finds the centroids
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// of the two classes of data.
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sample_type class1_center;
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sample_type class2_center;
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for (unsigned long i = 0; i < projected_samples.size(); ++i)
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{
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if (labels[i] == 1)
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class1_center += projected_samples[i];
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else
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class2_center += projected_samples[i];
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}
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const int points_per_class = samples.size()/2;
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class1_center /= points_per_class;
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class2_center /= points_per_class;
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// Now classify points by which center they are nearest. Recall that the data
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// is made up of two concentric circles. Normally you can't separate two concentric
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// circles by checking which points are nearest to each center since they have the same
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// centers. However, the kernel trick makes the data separable and the loop below will
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// perfectly classify each data point.
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for (unsigned long i = 0; i < projected_samples.size(); ++i)
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{
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double distance_to_class1 = length(projected_samples[i] - class1_center);
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double distance_to_class2 = length(projected_samples[i] - class2_center);
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bool predicted_as_class_1 = (distance_to_class1 < distance_to_class2);
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// Now print a message for any misclassified points.
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if (predicted_as_class_1 == true && labels[i] != 1)
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cout << "A point was misclassified" << endl;
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if (predicted_as_class_1 == false && labels[i] != 2)
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cout << "A point was misclassified" << endl;
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}
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// Next, note that classifying a point based on its distance between two other
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// points is the same thing as using the plane that lies between those two points
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// as a decision boundary. So let's compute that decision plane and use it to classify
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// all the points.
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sample_type plane_normal_vector = class1_center - class2_center;
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// The point right in the center of our two classes should be on the deciding plane, not
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// on one side or the other. This consideration brings us to the formula for the bias.
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double bias = dot((class1_center+class2_center)/2, plane_normal_vector);
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// Now classify points by which side of the plane they are on.
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for (unsigned long i = 0; i < projected_samples.size(); ++i)
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{
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double side = dot(plane_normal_vector, projected_samples[i]) - bias;
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bool predicted_as_class_1 = (side > 0);
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// Now print a message for any misclassified points.
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if (predicted_as_class_1 == true && labels[i] != 1)
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cout << "A point was misclassified" << endl;
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if (predicted_as_class_1 == false && labels[i] != 2)
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cout << "A point was misclassified" << endl;
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}
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// It would be nice to convert this decision rule into a normal decision_function object and
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// dispense with the empirical_kernel_map. Happily, it is possible to do so. Consider the
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// following example code:
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decision_function<kernel_type> dec_funct = ekm.convert_to_decision_function(plane_normal_vector);
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// The dec_funct now computes dot products between plane_normal_vector and the projection
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// of any sample point given to it. All that remains is to account for the bias.
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dec_funct.b = bias;
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// now classify points by which side of the plane they are on.
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for (unsigned long i = 0; i < samples.size(); ++i)
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{
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double side = dec_funct(samples[i]);
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// And let's just check that the dec_funct really does compute the same thing as the previous equation.
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double side_alternate_equation = dot(plane_normal_vector, projected_samples[i]) - bias;
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if (abs(side-side_alternate_equation) > 1e-14)
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cout << "dec_funct error: " << abs(side-side_alternate_equation) << endl;
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bool predicted_as_class_1 = (side > 0);
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// Now print a message for any misclassified points.
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if (predicted_as_class_1 == true && labels[i] != 1)
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cout << "A point was misclassified" << endl;
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if (predicted_as_class_1 == false && labels[i] != 2)
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cout << "A point was misclassified" << endl;
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}
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}
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// ----------------------------------------------------------------------------------------
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void generate_concentric_circles (
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std::vector<sample_type>& samples,
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std::vector<double>& labels,
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const int num
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)
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{
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sample_type m(2,1);
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samples.clear();
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labels.clear();
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dlib::rand rnd;
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// make some samples near the origin
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double radius = 1.0;
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for (long i = 0; i < num; ++i)
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{
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double sign = 1;
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if (rnd.get_random_double() < 0.5)
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sign = -1;
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m(0) = 2*radius*rnd.get_random_double()-radius;
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m(1) = sign*sqrt(radius*radius - m(0)*m(0));
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samples.push_back(m);
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labels.push_back(1);
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}
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// make some samples in a circle around the origin but far away
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radius = 5.0;
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for (long i = 0; i < num; ++i)
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{
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double sign = 1;
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if (rnd.get_random_double() < 0.5)
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sign = -1;
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m(0) = 2*radius*rnd.get_random_double()-radius;
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m(1) = sign*sqrt(radius*radius - m(0)*m(0));
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samples.push_back(m);
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labels.push_back(2);
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}
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}
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// ----------------------------------------------------------------------------------------
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