How Does the Kernel Trick Work?

The kernel trick relies on the inner products of vectors. For SVMs, the decision function is based on the dot products of vectors within the input space. Kernel functions replace these dot products with a non-linear function that computes a dot product in a higher-dimensional space. Importantly, the computation of this dot product via the kernel function does not require explicit knowledge of the coordinates in the higher space, thus saving computational resources and time.

The kernel trick is typically expressed as:

K(x,y)=ϕ(x)⋅ϕ(y)

where,

x and y are two vectors in the original input space
\phi
is the mapping function to the higher-dimensional space.

Kernel Trick in Support Vector Classification

Support Vector Machines (SVMs) have proven to be a powerful and versatile tool for classification tasks. A key component that significantly enhances the capabilities of SVMs, particularly in dealing with non-linear data, is the Kernel Trick. This article delves into the intricacies of the Kernel Trick, its motivation, implementation, and practical applications.

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