SVM

SVM Code

Introduction

Support Vector Machines (SVMs) are a highly versatile and powerful class of supervised machine learning algorithms designed for tasks such as classification and, to a lesser extent, regression. Their popularity stems from their ability to handle both linearly separable and non-linearly separable data with remarkable efficiency, making them one of the most effective tools in a data scientist’s toolkit.

At the core of SVMs is the principle of finding the optimal hyperplane that separates different classes of data points. This hyperplane serves as a decision boundary that distinctly classifies data points into predefined categories. The unique strength of SVM lies in its focus on maximizing the margin, the distance between the hyperplane and the nearest data points from each class, known as support vectors. By relying on this maximum margin principle, SVMs achieve excellent generalization, allowing them to make accurate predictions on new, unseen data.

Handling Linearly Separable Data

SVMs can handle both linearly separable and non-linearly separable data by using different kernel functions. In the case of linearly separable data, the SVM algorithm finds the optimal hyperplane that maximizes the margin between the classes. This is done by solving an optimization problem that minimizes the norm of the weight vector (the vector perpendicular to the hyperplane) while ensuring that all data points are correctly classified and lie on the correct side of the hyperplane.

Handling Non-Linearly Separable Data

For non-linearly separable data, the kernel trick is used to transform the data into a higher-dimensional feature space where it becomes linearly separable. The kernel function maps the input data from the original input space to a higher-dimensional feature space, where a linear hyperplane can separate the classes effectively.

Linear Separators: At their core, SVMs are linear separators. They work by finding a hyperplane in an N-dimensional space (N being the number of features) that distinctly classifies the data points. The optimal hyperplane is the one that has the maximum margin, which is the maximum distance between data points of different classes.

Kernel Trick and Dot Product: Often, data isn't linearly separable in its original form. SVMs employ kernels to solve this by transforming data into a higher-dimensional space where a hyperplane can be used to separate classes linearly. The kernel function computes the dot product of the vectors in this higher-dimensional space. The dot product is critical as it measures the angle and distance between vectors, helping to determine the separation margin.

Common Kernel Functions:

Polynomial Kernel: This kernel transforms the data by taking the dot product of the vectors and raising it to the power of 𝑑 (degree). It is represented as 𝐾(𝑥,𝑥′)=(1+𝑥⋅𝑥′)𝑑. For example, with 𝑑=2, it squares the dot product plus one.
Radial Basis Function (RBF) Kernel: The RBF or Gaussian kernel, defined as 𝐾(𝑥,𝑥′)=exp(−𝛾‖𝑥−𝑥′‖^2), focuses on the distance between the vectors. Here, 𝛾 is a parameter that defines how much influence a single training example has.

Example with Polynomial Kernel: Consider a two-dimensional point 𝑥=(𝑥1,𝑥2). Using a polynomial kernel with 𝑟=1 and 𝑑=2, the kernel function would transform this point into a higher-dimensional space: 𝐾(𝑥,𝑥)=(1+𝑥1𝑥1+𝑥2𝑥2)^2 This increases the dimensions to include terms like 𝑥1^2,x2^2, and x1x2, among others, effectively "casting" the original point into a space where linear separation might be more feasible.

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