Tausif Tech Hub

 

Kernel Functions in SVM: Transforming Non-Linear Models to Linear Models

Support Vector Machines (SVM) work well for linear classification, but real-world data is often non-linearly separable. Kernel functions help transform such non-linear data into a higher-dimensional space where it becomes linearly separable.

How Kernels Work

1. Feature Mapping: A kernel function implicitly maps input data $x$ from a lower-dimensional space to a higher-dimensional feature space where linear separation is possible.

2. Avoids Explicit Computation: Instead of computing the transformation explicitly, kernels calculate the dot product in the higher-dimensional space efficiently using:

   $$K(x_i, x_j) = \phi(x_i) \cdot \phi(x_j)$$

   where $\phi(x)$ is the transformation function.

Common Kernel Functions

1. Linear Kernel: $K(x_i, x_j) = x_i \cdot x_j$

   • Used when data is already linearly separable.

2. Polynomial Kernel: $K(x_i, x_j) = (x_i \cdot x_j + c)^d$

   • Captures non-linear relationships with polynomial degree $d$.

3. Radial Basis Function (RBF) Kernel: $K(x_i, x_j) = e^{-\gamma ||x_i - x_j||^2}$

   • Maps data into an infinite-dimensional space, handling highly non-linear structures.

4. Sigmoid Kernel: $K(x_i, x_j) = \tanh(\alpha x_i \cdot x_j + c)$

   • Inspired by neural networks but less commonly used.

Advantages of Kernels in SVM

• Handles Complex Patterns: Makes SVM effective for non-linearly separable data.

• Computational Efficiency: Avoids explicit transformation, reducing computational cost.

• Wide Applicability: Used in image recognition, bioinformatics, and text classification.

By using kernel tricks, SVM transforms non-linear problems into a solvable linear problem, enhancing its classification power.