What Are Kernel Functions? Learn Their Role in Machine Learning

What is the Kernel function?

Kernel is a technique in which we expand our data points to a higher dimension, allowing us to draw a clear boundary or distinguish between different classes.

Kernel function is a mathematical function that enables algorithms to operate in a high-dimensional feature space without explicitly computing the coordinates of the data in that space. This is known as the kernel trick.

1. Purpose:

   • Kernels allow algorithms (like Support Vector Machines (SVM)) to learn nonlinear relationships by implicitly mapping input data into higher-dimensional spaces.

   • They avoid the computational cost of explicitly transforming the data.

2. Mathematical Definition:

   • A kernel function K(x,x′) computes the dot product of vectors ϕ(x) and ϕ(x′) in some feature space:

     K(x,x′)=⟨ϕ(x),ϕ(x′)⟩

   • Here, ϕ is a feature mapping from input space to a higher-dimensional space.

3. Common Kernel Functions:

   • Linear Kernel:

     K(x,x′)=x^T * x′

     (No transformation, works for linearly separable data.)

   • Polynomial Kernel:

     K(x,x′)=(x^T * x′+c)d

     (Captures polynomial relationships.)

What if there, nothing like a kernel?

• Many machine learning algorithms (like SVM, linear regression) work well only if the data is linearly separable.

• If data is nonlinear (e.g., spirals, circles), we need to transform it into a higher-dimensional space where it becomes separable.

• Enables nonlinear classification/regression without explicitly computing high-dimensional transformations.

• Reduces computational complexity (since ϕ(x) is never calculated directly).

• Works in SVMs, Kernel PCA, Gaussian Processes, etc.

Example:

Suppose we have two classes: Class A: +1,-1, and Class B: 0.

Any line (e.g., x>0.5) will misclassify one of the points.

In 1D, a "line" is just a threshold value (e.g., x>c or x<c ).
We try different thresholds to separate Class A and Class B.

Attempt 1: Threshold at x>0.5

• Rule: If x>0.5, predict Class A; else, predict Class B.

• Results:

  • x=+1 → 1>0.51 → Correct (A)

  • x=0 → 0>0.50 → Correct (B)

  • x=−1 → −1>0.5 → Correct (B)

• Problem:

  • x=−1 is misclassified as B (but it’s actually A).

Attempt 2: Threshold at x<−0.5

• Rule: If x<−0.5, predict A; else, predict B.

• Results:

  • x=−1 → −1<−0.5 → Correct (A)

  • x=0 → 0<−0.50 → Correct (B)

  • x=+1 → +1<−0.5 → Correct (B)

• Problem:

  • x=+1 is misclassified as B (but it’s actually A).

Attempt 3: Threshold at x>0

• Rule: If x>0, predict A; else, predict B.

• Results:

  • x=+1 → 1>0 → Correct (A)

  • x=0 → 0>0 → Correct (B)

  • x=−1 → −1>0 → Correct (B)

• Problem:

  • In 1D, no single threshold (line) can separate A and B because Class A surrounds Class B. x=−1 is misclassified as B (but it’s A).

In 1D, no single threshold (line) can separate A and B because Class A surrounds Class B.

How Lifting to 2D Solves the Problem

By mapping x to (x,x²), we add a new dimension:

• Class A:

  • (−1,1)

  • (+1,1)

• Class B:

  • (0,0)

Now, a horizontal line at y=0.5 separates them:

• All A points are above the line (y=1>0.5).

• The B point is below the line (y = 0 < 0.5).

• The new dimension (x²) "lifts" Class A up, making them separable by a horizontal line.

This is why kernels are powerful—they implicitly lift data to higher dimensions without explicitly computing the transformation!

Summary

Kernel functions enable algorithms to operate in high-dimensional feature spaces without explicitly computing data coordinates there, using the kernel trick. This allows for nonlinear classification and regression, avoiding direct high-dimensional computations, thus reducing complexity. They are useful in algorithms like SVMs, kernel PCA, and Gaussian processes. Common kernels include linear, polynomial, Gaussian RBF, and sigmoid. By mapping data to higher dimensions, kernels can effectively separate classes that are inseparable in lower dimensions, exemplified by mapping 1D data to 2D for clearer separability.