SVM Kernel Function

In the last article, we saw about the SVM algorithm, its working, and its applications. However, in those articles, we just briefly mentioned one of the most important factors behind the working of the SVM- kernel function. Kernel functions play a fundamental role in the smooth working of the SVM algorithm. We can certainly say that the kernel is the most crucial step in the working of the SVM algorithm since it determines the form of output that we desire.

PythonGeeks brings to you, an article that talks about the functionality of the kernel function. We will discuss what actually is a kernel function, the working of the kernel function, and the rules associated with it. In the end, we will go through the various types of kernel functions that we can choose according to the classification problem that we have to tackle. So, let’s dive straight into the introduction of the kernel function.

Introduction to SVM Kernel Function

Simply defined, the kernel is a function that we use in SVM to get the desired output. The kernel performs the task of accepting the input from the user and transforming it into the desired output. The kernel provides the programmer with pre-defined structures of mathematical functions which helps them to surpass complexities in calculations.

As we have seen in the SVM tutorial, it becomes difficult for us to decide the decision boundary when the dimensions of the data representation plane are higher. The most amazing feature of the kernel is that it works efficiently even on datasets having higher dimensions. It can perform complex calculations of the higher dimension datasets in a quite simple manner.

Owing to its smooth functionality with handling higher dimension datasets, kernels can handle datasets with dimensionality up to infinite. Kernel functions can smoothly help the algorithm in deciding the hyperplane without escalating the complexities of the dataset. Depending on the tasks they have to perform, different algorithms can choose from the different kernel functions to serve their purpose.

The most widely used type of kernel function is Radial Basis Function (RBF) since it has localized and finite number response along the entire x-axis.

The kernel functions return the inner product between two points in suitable feature space as the output for the smooth classification process. Thus it alters a notion of similarity, with a very low computational cost even in very high-dimensional spaces.

Now that we know what exactly one conveys with the term “kernel”, let us look at how this kernel function works. We will also cover topics like kernel trick and overfitting.

Working of SVM Kernel Function

Kernels tend to solve a non-linearly separable problem with the help of linear classifying features. This technique in SVM is known as the kernel trick of the SVM. We use the kernel functions as parameters of the SVM algorithm. They are efficient in deciding the dimensions of the hyperplane and thus effectively decide the decision boundary.

We can modify the value of the kernel function in the SVM algorithm. Kernel functions can attain their value from a wide range of options like polynomial, linear, and so on. If we consider the working of the linear kernel function, then the dataset would be two-dimensional with a linear decision boundary. Such types of kernel functions can efficiently convey the decision boundary for higher dimensions datasets as well.

Owing to the structures of the kernel function, we do not need to perform complex calculations for the classification. Kernel function takes care of all these complexities. We just need to select the appropriate type of kernel function as the input and it will help the algorithm to classify the dataset. We can even solve the overfitting problem of the SVM using the kernel function.

Overfitting problem occurs when the dataset has more feature sets as compared to the sample sets of the data. We can avoid the overfitting in SVM by either expanding the dataset or by choosing the appropriate kernel function.

The most common type of kernel function that is widely used is the RBF kernel since it is efficient with larger as well as smaller datasets. However, due to the universality of the RBF, it may cause overfitting while performing on smaller datasets.

Because of this reason, we should prefer kernels like linear and polynomial kernels to handle smaller datasets.

Hence, we have covered the working of the kernel functions. Now, let us look at the rules of the kernel function.

Rules of SVM Kernel Functions

There are a set of predefined rules that one must follow in order for the smooth working of the kernel in the SVM algorithm. These rules play an important role in deciding the type of kernel function we have to use for our algorithm for classification. One of such rules of the kernel function is the window function or the moving window classifier. The function is defined as:

K (x)= 1 if |x| ≤ 1 or 0 in all other cases

The function takes the value 1 inside the closed surface of radius 1 whose center lies at the origin. It takes 0 in all other cases of the dataset.

Here, the width of the window function is predefined. This helps the function to decide on the type of kernel we have to use.

Types of Kernel Functions in SVM

Let us have a quick look at the most used types of kernel functions that we can use for the classification of the dataset.

1. Polynomial Kernel Function

This type of kernel is mostly used for image processing algorithms. It is a general representation of the kernels having degree more than one. The polynomial kernel is further divided into two types:

a. Homogeneous Polynomial Kernel

K(x_i, x_j) = (xi . xj)^d

In this function, we take the dot product of both the numbers and d represents the degree of the polynomial.

b. Inhomogeneous Polynomial Kernel

K (xi, xj) = (xi . xj + c)^d here c is a constant

2. Gaussian RBF Kernel

RBL is the acronym for Radial Basis Function. We prefer this kernel function when we do not have any prior knowledge of the data.

K (xi, xj) = exp(-ϒ||xi – xj||)²

3. Sigmoid Kernel Function

We prefer this type of kernel function in the case of neural networks. The mathematical representation of the sigmoid kernel function is

K (xi, xj) = tanh(α x^ay +c)

4. Hyperbolic Tangent Kernel Function

This type of kernel is useful when we have to deal with neural networks. The mathematical representation of the function is

K (xi, xj) = tanh(kxi . xj +c)

5. Linear Kernel Function

This is the most basic type of kernel that we use for SVM classification. It is a one-dimensional kernel. The mathematical representation is

K (xi, xj) = xi. xj +c

6. Graph Kernel Function

We prefer this kernel function for the inner part of the graph. It is useful for measuring the similarity between the pairs of graphs. They are useful in the areas like bioinformatics, chemoinformatics, and so on.

7. String Kernel Function

The basis of the working of this kernel is the strings we pass as input. The main use of this kernel is in the field of text classification. They are also used in areas like text mining, genome analysis, etc.

8. Tree Kernel Function

This kernel is associated with problems related to the tree structure. The main aim of this kernel is to split the data in a tree structure which helps the SVM in classifying the data points. This kernel is useful for problems like language classification and areas like Natural Language Processing (NLP).

9. Bessel Function of First Kind Kernel

We can use the Bessel Function to remove the cross term in the mathematical functions.

k (x,y)= Jv+1(σ II x-y II) / II x-y II -n(v+1)

10. Laplace RBF Kernel

This type of kernel is a general-purpose kernel that we prefer when there is no prior knowledge of the data.

K(x,y) = exp(- || x-y|| / µ)

11. ANOVA radial basis Kernel

K (x,y) = Σ exp (-σ (x_k – y_k)²)^d

12. Linear Splines kernel in one-dimension

This type of kernel is useful when we have to deal with a large dataset containing sparse data vectors. We use this kernel in text categorization. It is efficient while dealing with regression.

K(x,y) = 1+ xy + xy min(x,y) – x+y/2 min(x,y)² + 1/3 min(x,y)³

Conclusion

With this, we have reached the end of this article about kernel function. We discussed the definition of the kernel, it’s working, and the types of the kernel function. We came to know about how kernel functions help in determining the type of kernel we prefer for the problems we deal with. Hope that this article from PythonGeeks was able to clear your understanding of the kernel function.