Skip to main content

Kernelized Support Vector Machines

Image
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

 

Categories:

Comments

Post a Comment

Popular posts from this blog

Relation of Model Complexity to Dataset Size

Dr. Rishabh Pathak
Core Concept The relationship between model complexity and dataset size is fundamental in supervised learning, affecting how well a model can learn and generalize. Model complexity refers to the capacity or flexibility of the model to fit a wide variety of functions. Dataset size refers to the number and diversity of training samples available for learning. Key Points 1. Larger Datasets Allow for More Complex Models When your dataset contains more varied data points , you can afford to use more complex models without overfitting. More data points mean more information and variety, enabling the model to learn detailed patterns without fitting noise. Quote from the book: "Relation of Model Complexity to Dataset Size. It’s important to note that model complexity is intimately tied to the variation of inputs contained in your training dataset: the larger variety of data points your dataset contains, the more complex a model you can use without overfitting....
Post a Comment
Read more

Linear Models

Dr. Rishabh Pathak
1. What are Linear Models? Linear models are a class of models that make predictions using a linear function of the input features. The prediction is computed as a weighted sum of the input features plus a bias term. They have been extensively studied over more than a century and remain widely used due to their simplicity, interpretability, and effectiveness in many scenarios. 2. Mathematical Formulation For regression , the general form of a linear model's prediction is: y^ ​ = w0 ​ x0 ​ + w1 ​ x1 ​ + … + wp ​ xp ​ + b where; y^ ​ is the predicted output, xi ​ is the i-th input feature, wi ​ is the learned weight coefficient for feature xi ​ , b is the intercept (bias term), p is the number of features. In vector form: y^ ​ = wTx + b where w = ( w0 ​ , w1 ​ , ... , wp ​ ) and x = ( x0 ​ , x1 ​ , ... , xp ​ ) . 3. Interpretation and Intuition The prediction is a linear combination of features — each feature contributes prop...
Post a Comment
Read more

Predicting Probabilities

Dr. Rishabh Pathak
1. What is Predicting Probabilities? The predict_proba method estimates the probability that a given input belongs to each class. It returns values in the range [0, 1] , representing the model's confidence as probabilities. The sum of predicted probabilities across all classes for a sample is always 1 (i.e., they form a valid probability distribution). 2. Output Shape of predict_proba For binary classification , the shape of the output is (n_samples, 2) : Column 0: Probability of the sample belonging to the negative class. Column 1: Probability of the sample belonging to the positive class. For multiclass classification , the shape is (n_samples, n_classes) , with each column corresponding to the probability of the sample belonging to that class. 3. Interpretation of predict_proba Output The probability reflects how confidently the model believes a data point belongs to each class. For example, in ...
Post a Comment
Read more

Supervised Learning

Dr. Rishabh Pathak
What is Supervised Learning? ·     Definition: Supervised learning involves training a model on a labeled dataset, where the input data (features) are paired with the correct output (labels). The model learns to map inputs to outputs and can predict labels for unseen input data. ·     Goal: To learn a function that generalizes well from training data to accurately predict labels for new data. ·          Types: ·          Classification: Predicting categorical labels (e.g., classifying iris flowers into species). ·          Regression: Predicting continuous values (e.g., predicting house prices). Key Concepts: ·     Generalization: The ability of a model to perform well on previously unseen data, not just the training data. ·         Overfitting and Underfitting: ·    ...
Post a Comment
Read more