kernelkmeans

Kernel k-means is a clustering algorithm that extends the standard k-means algorithm by operating in a high-dimensional feature space. Instead of directly computing distances in this feature space, which can be computationally expensive or even infinite-dimensional, kernel k-means uses the kernel trick. This allows it to implicitly map data points to this feature space and compute distances or similarities without explicitly performing the mapping.

The core idea is to redefine the cluster centroids and the distance metric. In standard k-means, centroids are represented by the mean of data points within a cluster. In kernel k-means, the centroids are represented implicitly, and the objective is to minimize the sum of squared distances between data points and their assigned cluster centroids in the feature space. This minimization is achieved by iteratively reassigning data points to clusters based on which implicit centroid minimizes the kernelized distance.

The choice of kernel function is crucial. Common choices include the polynomial kernel and the radial basis function (RBF) kernel. The kernel function determines the nature of the similarity or distance measure in the feature space, influencing the shape and separability of the clusters. Kernel k-means can discover non-linear cluster structures that are not detectable by standard k-means. However, like standard k-means, it requires the number of clusters to be pre-specified. The algorithm's computational complexity can be higher than standard k-means, especially with complex kernel functions or large datasets.