Kernel method

What is Kernel method in machine learning?

Kernel methods are types of algorithms that are used for pattern analysis. These methods involve using linear classifiers to solve nonlinear problems. Essentially, kernel methods are algorithms that make it possible to implicitly project the data in a high-dimensional space. The utilization of kernel functions for the purpose of making computations was first introduced in 1964. A couple of decades later, multiple authors proposed a neural network, ‘radial basis function (RBF)’, which was based on the kernel functions that were very popularly used in several applicative fields. From 1995, kernel methods have earned a fundamental place in machine learning when support vector machines (SVMs) were proposed. SVMs have performed better than other machine learning algorithms in multiple applications.

Kernels are used in Support Vector Machines (SVMs) to solve regression and classification problems. Support Vector Machines use the Kernel Trick to transform linearly inseparable data into linearly separable data, thus finding an optimal boundary for possible outputs.

You can break SVM strategy down into two steps: First, the data is projected implicitly onto a high-dimensional space through the kernel trick. The second step involves applying a linear classifier to the projected data. Because the linear classifier can solve a very limited class of problems, the kernel trick is employed to empower the linear classifier, enabling the SVM to solve a larger class of problems.

Support Vector Machines make use of the kernel method to use data as input and transform into the required type of processing data. “Kernel” is used because of the set of mathematical functions used in Support Vector Machine that gives the window to manipulate the data. The Kernel function will usually convert the training set of data so that a non-linear decision surface can be transformed to a linear equation in a higher number of dimension spaces. Essentially, it gives back the inner product between two points in a standard feature dimension.

Kernel functions are applied to every data instance for the purpose of mapping the original nonlinear observations into a higher-dimensional space. These observations become separable in the higher-dimensional space.

What are the types of Kernel methods in SVM models?

Support vector machines use various kinds of kernel methods. Here are a few of them:

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1. Linear Kernel

If there are two kernels named x1 and x2, the linear kernel can be defined by the dot product of the two vectors:

K(x1, x2) = x1 . x2

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2. Polynomial Kernel

We can define a polynomial kernel with this equation:

K(x1, x2) = (x1 . x2 + 1)d

Here, x1 and x2 are vectors and d represents the degree of the polynomial.

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3. Gaussian Kernel

The Gaussian kernel is an example of a radial basis function kernel. It can be represented with this equation:

k(xi, xj) = exp(-𝛾||xi - xj||2)

The given sigma has a vital role in the performance of the Gaussian kernel. It should be carefully tuned according to the problem, neither overestimated and nor underestimated.

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4. Exponential Kernel

Exponential kernels are closely related to Gaussian kernels. These are also radial basis kernel functions. The difference between these two types of kernels is that the square of the norm is removed in Exponential kernels.

The function of an exponential function is:

k(x, y) =exp(-||x -y||22)

5. Laplacian Kernel

A Laplacian kernel is less prone to changes. It is equal to an exponential kernel.

The equation of a Laplacian kernel is

k(x, y) = exp(- ||x - y||)

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6. Hyperbolic or the Sigmoid Kernel

Hyperbolic or Sigmoid kernels are used in neural networks. These kernels use a bipolar sigmoid activation function.

The hyperbolic kernel can be represented with this equation:

k(x, y) = tanh(xTy + c)

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7. Anova radial basis kernel

This is another type of radial basis kernel function. Anova radial basis kernels work rather well in multidimensional regression problems.

An Anova radial basis kernel can be represented with this equation:

k(x, y) = k=1nexp(-(xk -yk)2)d

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What is kernel PCA in machine learning?

Principal component analysis is a tool that is used to reduce the dimension of the data. It enables you to reduce the dimension of the data without loosing too much information. Principal Component Analysis reduces the dimension by identifying a few orthogonal linear combinations (principal components) of the original variables that have the largest variance.

The first principal component captures the majority of the variance in the data. The following principal component is orthogonal to the first principal component. It captures the remaining variance that is left of first principal component and so on. There are as many principal components as there are original variables. These principal components tend to be uncorrelated and are ordered in such a manner that the first several principal components will explain the majority of the variance of the original data.

But Principal Component Analysis happens to be a linear method. You can only apply it to datasets which are linearly separable. It does an excellent job for such datasets. But if you use it on non-linear datasets, you might end up with a result which may not be the optimal dimensionality reduction.

Kernal PCA (Kernal Principal Component Analysis) employs a kernel function to project dataset into a higher dimensional feature space in which it is linearly separable. This is quite similar to the concept of support vector machines. The data gets mapped to a higher dimensional space, but turns out to be on a lower-dimensional subspace of it. You increase the dimensionality in order for you to be able to decrease it. Essentially, with the kernel trick, you do not actually need to explicitly consider the higher-dimensional space, which means that the leap in dimensionality that could be rather confusing basically gets performed undercover.