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Kernelized Support Vector Machines

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Dr. Rishabh Pathak

1. Introduction to SVMs

  • Support Vector Machines (SVMs) are supervised learning algorithms primarily used for classification (and regression with SVR).
  • They aim to find the optimal separating hyperplane that maximizes the margin between classes for linearly separable data.
  • Basic (linear) SVMs operate in the original feature space, producing linear decision boundaries.

2. Limitations of Linear SVMs

  • Linear SVMs have limited flexibility as their decision boundaries are hyperplanes.
  • Many real-world problems require more complex, non-linear decision boundaries that linear SVM cannot provide.

3. Kernel Trick: Overcoming Non-linearity

  • To allow non-linear decision boundaries, SVMs exploit the kernel trick.
  • The kernel trick implicitly maps input data into a higher-dimensional feature space where linear separation might be possible, without explicitly performing the costly mapping.

How the Kernel Trick Works:

  • Instead of computing the coordinates of data points in high-dimensional space (which could be infinite-dimensional), SVM calculates inner products (similarity measures) directly using kernel functions.
  • These inner products correspond to an implicit mapping into the higher-dimensional space.
  • This avoids the curse of dimensionality and reduces computational cost.

4. Types of Kernels

The most common kernels:

1.      Polynomial Kernel

  • Computes all polynomial combinations of features up to a specified degree.
  • Enables capturing interactions and higher-order feature terms.
  • Example: kernel corresponds to sums like feature1², feature1 × feature2⁵, etc..

2.     Radial Basis Function (RBF) Kernel (Gaussian Kernel)

  • Corresponds to an infinite-dimensional feature space.
  • Measures similarity based on the distance between points in original space, decreasing exponentially with distance.
  • Suitable when relationships are highly non-linear and not well captured by polynomial terms.

5. Important Parameters in Kernelized SVMs

1.      Regularization parameter (C)

  • Controls the trade-off between maximizing the margin and minimizing classification error.
  • A small C encourages a wider margin but allows some misclassifications (more regularization).
  • A large C tries to classify all training points correctly but might overfit.

2.     Kernel choice

  • Selecting the appropriate kernel function is critical (polynomial, RBF, linear, etc.).
  • The choice depends on the data and problem structure.

3.     Kernel-specific parameters

  • Each kernel function has parameters:
  • Polynomial kernel: degree of polynomial.
  • RBF kernel: gamma (shape of Gaussian; higher gamma means points closer).
  • These parameters govern the flexibility and complexity of the decision boundary.

6. Strengths and Weaknesses

Strengths

  • Flexibility:
  • SVMs can create complex, non-linear boundaries suitable for both low and high-dimensional data,.
  • Effective in high dimensions:
  • Works well even if the number of features exceeds the number of samples.
  • Kernel trick:
  • Avoids explicit computations in very high-dimensional spaces, saving computational resources.

Weaknesses

  • Scalability:
  • SVMs scale poorly with the number of samples.
  • Practical for datasets up to ~10,000 samples; larger datasets increase runtime and memory significantly.
  • Parameter tuning and preprocessing:
  • Requires careful preprocessing (feature scaling is important), tuning of C, kernel, and kernel-specific parameters for good performance.
  • Interpretability:
  • Model is difficult to interpret; explaining why a prediction was made is challenging.

7. When to Use Kernelized SVMs?

  • Consider kernelized SVMs if:
  • Your features have similar scales or represent homogeneous measurements (e.g., pixel intensities).
  • The dataset is not too large (under ~10,000 samples).
  • You require powerful non-linear classification with well-separated classes.

8. Mathematical Background (Overview)

  • The underlying math is involved and detailed in advanced texts such as The Elements of Statistical Learning by Hastie, Tibshirani, and Friedman.
  • Conceptually:
  • The primal optimization problem tries to maximize the margin while penalizing misclassifications.
  • The dual problem allows the introduction of kernels, enabling use of the kernel trick.

Summary

Aspect

Details

Purpose

Classification with linear or non-linear decision boundaries

Key idea

Map data to higher-dimensional space via kernels (kernel trick)

Common kernels

Polynomial, RBF (Gaussian)

Parameters

Regularization C, kernel type, kernel-specific params (degree, gamma)

Strengths

Flexible decision boundaries, works well in high-dimensions

Weaknesses

Poor scaling to large datasets, requires tuning, less interpretable

Use cases

Data with uniform feature scaling, moderate size datasets

 

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