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204.6.5 The Non-Linear Decision Boundary

Link to the previous post : https://statinfer.com/204-6-4-building-svm-model-in-python/

The Non-Linear Decision Boundary

  • In the above examples we can clearly see the decision boundary is linear.
  • SVM works well when the data points are linearly separable.
  • If the decision boundary is non-liner then SVM may struggle to classify.
  • Observe the below examples, the classes are not linearly separable.
  • SVM has no direct theory to set the non-liner decision boundary models.

Mapping to Higher Dimensional Space

  • The original maximum-margin hyperplane algorithm proposed by Vapnik in 1963 constructed a linear classifier.
  • To fit a non liner boundary classier, we can create new variables(dimensions) in the data and see whether the decision boundary is linear.
  • In 1992, Bernhard E. Boser, Isabelle M. Guyon and Vladimir N. Vapnik suggested a way to create nonlinear classifiers by applying the kernel trick.
  • In the below example, A single linear classifier is not sufficient.
  • Lets create a new variable x2=(x1)2. In the higher dimensional space.
  • We can clearly see a possibility of single linear decision boundary.
  • This is called kernel trick.

Kernel Trick


  • We used a function ϕ(x)=(x,(x2)) to transform the data x into a higher dimensional space.
  • In the higher dimensional space, we could easily fit a liner decision boundary.
  • This function ϕ(x) is known as kernel function and this process is known as kernel trick in SVM.
  • Kernel trick solves the non-linear decision boundary problem much like the hidden layers in neural networks.
  • Kernel trick is simply increasing the number of dimensions. It is to make the non-linear decision boundary in lower dimensional space as a linear decision boundary, in higher dimensional space.
  • In simple words, Kernel trick makes the non-linear decision boundary to linear (in higher dimensional space).

Kernel Function Examples

Name Function Type problem
Polynomial Kernel \(x_i^t x_j +1)^q\ q is degree of polynomial Best for Image processing
Sigmoid Kernel \tanh(ax_i^t x_j +k)\ k is offset value Very similar to neural network
Gaussian Kernel \e^(||x_i - x_j||^2/2 \sigma^2)\ No prior knowledge on data
Linear Kernel \(1+x_i x_j min(x_i , x_j) - \frac{(x_i + x_j)}{2} min(x_i , x_j)^2 + \frac{min(x_i , x_j)^3}{3}\) Text Classification
Laplace Radial Basis Function (RBF) \(e^(-\lambda ||x_i - x_j||) , \lambda >= 0\) No prior knowledge on data
    • There are many more kernel functions.

Choosing the Kernel Function

  • Probably the most tricky part of using SVM.
  • The kernel function is important because it creates the kernel matrix, which summarizes all the data.
  • There is no proven theory for choosing a kernel function for any given problem. Still there is lot of research going on.
  • In practice, a low degree polynomial kernel or RBF kernel with a reasonable width is a good initial try.
  • Choosing Kernel function is similar to choosing number of hidden layers in neural networks. Both of them have no proven theory to arrive at a standard value.
  • As a first step, we can choose low degree polynomial or radial basis function or one of those from the list.

The next post is a practice session on kernel non-linear classifier.

Link to the next post : https://statinfer.com/204-6-6-practice-kernel-non-linear-classifier/

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