Radial Basis Function (RBF) Kernel: The Go-To Kernel

Radial Basis Function (RBF) Kernel: The Go-To Kernel

In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification.

You’re working on a Machine Learning algorithm like Support Vector Machines for non-linear datasets and you can’t seem to figure out the right feature transform or the right kernel to use. Well, fear not because Radial Basis Function (RBF) Kernel is your savior.

RBF kernels are the most generalized form of kernelization and is one of the most widely used kernels due to its similarity to the Gaussian distribution. The RBF kernel function for two points X₁ and X₂ computes the similarity or how close they are to each other. This kernel can be mathematically represented as follows:

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where,

1. ‘σ’ is the variance and our hyperparameter

2. ||X₁ - X₂|| _is the Euclidean (L_₂-norm) Distance between two points X₁ and X₂

Let d₁₂ be the distance between the two points X₁ and X₂, we can now represent d₁₂ as follows:

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Fig 2: Distance between two points in space [Image by Author]

The kernel equation can be re-written as follows:

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The maximum value that the RBF kernel can be is 1 and occurs when d₁₂ is 0 which is when the points are the same, i.e. X₁ = X₂.

  1. When the points are the same, there is no distance between them and therefore they are extremely similar
  2. When the points are separated by a large distance, then the kernel value is less than 1 and close to 0 which would mean that the points are dissimilar

Distance can be thought of as an equivalent to dissimilarity because we can notice that when distance between the points increases, they are less similar.

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