How is the transformation from the original feature set to the new space performed in SVM with kernels?
The transformation from the original feature set to the new space in Support Vector Machines (SVM) with kernels is a important step in the classification process. Kernels play a fundamental role in SVMs as they enable the algorithm to operate in a higher-dimensional feature space, where the data might be more separable. This transformation is performed using a technique called the kernel trick.
To understand the transformation process, let's first revisit the basic concept of SVMs. SVMs are binary classifiers that aim to find the optimal hyperplane that separates two classes of data points. In the original feature space, this hyperplane is a linear decision boundary. However, in some cases, the data may not be linearly separable, making it difficult for a linear classifier to accurately classify the data.
Kernels provide a way to address this limitation by implicitly mapping the data points from the original feature space to a higher-dimensional space, where the classes might become linearly separable. This mapping is achieved by applying a kernel function to the original feature vectors.
A kernel function takes two input vectors and computes their similarity or distance in the higher-dimensional feature space. The most commonly used kernel functions are the linear kernel, polynomial kernel, radial basis function (RBF) kernel, and sigmoid kernel. Each kernel function has its own characteristics and is suitable for different types of data.
Let's take a closer look at the transformation process for the RBF kernel, which is widely used due to its flexibility. Given a data point x in the original feature space, the RBF kernel transforms it into a new feature vector φ(x) in the higher-dimensional space. The RBF kernel is defined as:
K(x, y) = exp(-γ ||x – y||^2)
where γ is a user-defined parameter that controls the width of the kernel and ||x – y||^2 represents the squared Euclidean distance between x and y.
By applying the RBF kernel to each pair of data points, SVM constructs a decision boundary in the new feature space. The transformed data points are then classified based on their positions relative to this decision boundary.
It's important to note that the transformation from the original feature space to the new space is done implicitly, without explicitly computing the coordinates of the transformed data points. This is known as the kernel trick, which allows SVM to operate in the higher-dimensional space without the need for explicitly representing the transformed data points. This is computationally efficient, as the dimensionality of the new space can be much higher than the original feature space.
The transformation from the original feature set to the new space in SVM with kernels is achieved by applying a kernel function to the original feature vectors. Kernels enable SVM to operate in a higher-dimensional feature space, where the classes might become linearly separable. The kernel trick allows SVM to implicitly perform this transformation without explicitly computing the coordinates of the transformed data points.
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