Kernelized Support Vector Machines

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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