**Prerequisite:** Support Vector Machines

**Definition of a hyperplane and SVM classifier:**

For a linearly separable dataset having n features (thereby needing n dimensions for representation), a hyperplane is basically an (n – 1) dimensional subspace used for separating the dataset into two sets, each set containing data points belonging to a different class. For example, for a dataset having two features X and Y (therefore lying in a 2-dimensional space), the separating hyperplane is a line (a 1-dimensional subspace). Similarly, for a dataset having 3-dimensions, we have a 2-dimensional separating hyperplane, and so on.

In machine learning, Support Vector Machine (SVM) is a non-probabilistic, linear, binary classifier used for classifying data by learning a hyperplane separating the data.

**Classifying a non-linearly separable dataset using a SVM – a linear classifier:**

As mentioned above SVM is a linear classifier which learns an (n – 1)-dimensional classifier for classification of data into two classes. However, it can be used for classifying a non-linear dataset. This can be done by projecting the dataset into a higher dimension in which it is linearly separable!

To get a better understanding, let’s consider circles dataset.

`# importing libraries ` `import` `numpy as np ` `import` `matplotlib.pyplot as plt ` `from` `sklearn.datasets ` `import` `make_circles ` `from` `mpl_toolkits.mplot3d ` `import` `Axes3D ` ` ` `# generating data ` `X, Y ` `=` `make_circles(n_samples ` `=` `500` `, noise ` `=` `0.02` `) ` ` ` `# visualizing data ` `plt.scatter(X[:, ` `0` `], X[:, ` `1` `], c ` `=` `Y, marker ` `=` `'.'` `) ` `plt.show() ` |

The dataset is clearly a non-linear dataset and consists of two features (say, X and Y).

In order to use SVM for classifying this data, introduce another feature Z = X^{2} + Y^{2} into the dataset. Thus, projecting the 2-dimensional data into 3-dimensional space. The first dimension representing the feature X, second representing Y and third representing Z (which, mathematically, is equal to the radius of the circle of which the point (x, y) is a part of). Now, clearly, for the data shown above, the ‘yellow’ data points belong to a circle of smaller radius and the ‘purple’ data points belong to a circle of larger radius. Thus, the data becomes linearly separable along the Z-axis.

`# adding a new dimension to X ` `X1 ` `=` `X[:, ` `0` `].reshape((` `-` `1` `, ` `1` `)) ` `X2 ` `=` `X[:, ` `1` `].reshape((` `-` `1` `, ` `1` `)) ` `X3 ` `=` `(X1` `*` `*` `2` `+` `X2` `*` `*` `2` `) ` `X ` `=` `np.hstack((X, X3)) ` ` ` `# visualizing data in higher dimension ` `fig ` `=` `plt.figure() ` `axes ` `=` `fig.add_subplot(` `111` `, projection ` `=` `'3d'` `) ` `axes.scatter(X1, X2, X1` `*` `*` `2` `+` `X2` `*` `*` `2` `, c ` `=` `Y, depthshade ` `=` `True` `) ` `plt.show() ` |

Now, we can use SVM (or, for that matter, any other linear classifier) to learn a 2-dimensional separating hyperplane. This is how the hyperplane would look like:

`# create support vector classifier using a linear kernel ` `from` `sklearn ` `import` `svm ` ` ` `svc ` `=` `svm.SVC(kernel ` `=` `'linear'` `) ` `svc.fit(X, Y) ` `w ` `=` `svc.coef_ ` `b ` `=` `svc.intercept_ ` ` ` `# plotting the separating hyperplane ` `x1 ` `=` `X[:, ` `0` `].reshape((` `-` `1` `, ` `1` `)) ` `x2 ` `=` `X[:, ` `1` `].reshape((` `-` `1` `, ` `1` `)) ` `x1, x2 ` `=` `np.meshgrid(x1, x2) ` `x3 ` `=` `-` `(w[` `0` `][` `0` `]` `*` `x1 ` `+` `w[` `0` `][` `1` `]` `*` `x2 ` `+` `b) ` `/` `w[` `0` `][` `2` `] ` ` ` `fig ` `=` `plt.figure() ` `axes2 ` `=` `fig.add_subplot(` `111` `, projection ` `=` `'3d'` `) ` `axes2.scatter(X1, X2, X1` `*` `*` `2` `+` `X2` `*` `*` `2` `, c ` `=` `Y, depthshade ` `=` `True` `) ` `axes1 ` `=` `fig.gca(projection ` `=` `'3d'` `) ` `axes1.plot_surface(x1, x2, x3, alpha ` `=` `0.01` `) ` `plt.show() ` |

Thus, using a linear classifier we can separate a non-linearly separable dataset.

**A brief introduction to kernels in machine learning:**

In machine learning, a trick known as “kernel trick” is used to learn a linear classifier to classify a non-linear dataset. It transforms the linearly inseparable data into a linearly separable one by projecting it into a higher dimension. A kernel function is applied on each data instance to map the original non-linear data points into some higher dimensional space in which they become linearly separable.