I usually rely on pre-built libraries like libsvm for support vector machines (SVMs), even though understanding the underlying principles can be challenging. However, I still try to grasp the core concepts and the reasons behind them. This article aims to provide a conceptual overview of SVMs in a clear and accessible way.
Support Vector Machines were first introduced by Corinna Cortes and Vapnik in 1995. They have shown great effectiveness in handling small sample sizes, nonlinear problems, and high-dimensional data, making them suitable for tasks such as pattern recognition and function approximation.
An SVM is essentially a classifier that operates using support vectors. The term "machine" here refers to a model or algorithm used for classification, rather than a literal machine.
So, what exactly are support vectors? During the training process, it's observed that only a subset of the data points—those closest to the decision boundary—determine the final classifier. These key points are called support vectors.
Imagine a two-dimensional plot where the black line represents the decision boundary. Points near this line, such as R, S, and G, are considered support vectors because they define the position and orientation of the line.
Like neural networks, SVMs are learning models, but unlike neural networks, they rely on mathematical optimization techniques rather than iterative adjustments.
To better understand SVMs, let's revisit logistic regression, which serves as a stepping stone. Logistic regression is used for binary classification, mapping the output of a linear function through a sigmoid function to produce a probability between 0 and 1.
The hypothesis function for logistic regression is:
Here, $ x $ is an n-dimensional feature vector, and $ g(z) $ is the logistic function. The function maps any real number to the range (0,1), representing the probability that the class label $ y = 1 $.
As shown in the graph, the logistic function smoothly transitions from 0 to 1, allowing us to interpret the output as a likelihood.
Once we have the hypothesis function $ h_\theta(x) $, we can classify new examples. If $ h_\theta(x) \geq 0.5 $, we predict $ y = 1 $; otherwise, $ y = 0 $.
Although the logistic function is used for mapping, the actual classification decision depends on $ \theta^T x $. The goal is to find parameters $ \theta $ such that $ \theta^T x \geq 0 $ for positive examples and $ \theta^T x \leq 0 $ for negative ones.
In contrast, SVMs focus more on the margin between classes. Instead of maximizing the distance for all points, SVMs aim to maximize the distance between the closest points from each class—these are the support vectors.
This leads to the concept of functional and geometric margins. Functional margin measures how confident the model is about its prediction, while geometric margin measures the actual distance from the decision boundary.
By maximizing the geometric margin, SVMs ensure that the decision boundary is as far as possible from the nearest data points, leading to better generalization and robustness.
The optimal margin classifier is formulated as a quadratic optimization problem with linear constraints. Solving this allows us to determine the best hyperplane that separates the classes effectively.
In summary, while logistic regression focuses on probabilistic classification, SVMs emphasize maximizing the margin between classes. This distinction makes SVMs particularly powerful in complex and high-dimensional spaces.
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