Rectified Euler k-means and Beyond

TL;DR

This paper introduces two rectified Euler k-means algorithms, REK1 and REK2, which ensure centroids lie on the data's mapped space, improving clustering accuracy and data structure representation.

Contribution

The paper proposes novel rectified Euler k-means methods that constrain centroids to the mapped space, addressing the deviation issue in traditional EulerK clustering.

Findings

REK1 and REK2 improve centroid placement on the mapped space.

Experiments show enhanced clustering performance with the proposed methods.

The methods are extendable to other kernel clustering problems.

Abstract

Euler k-means (EulerK) first maps data onto the unit hyper-sphere surface of equi-dimensional space via a complex mapping which induces the robust Euler kernel and next employs the popular $k$-means. Consequently, besides enjoying the virtues of k-means such as simplicity and scalability to large data sets, EulerK is also robust to noises and outliers. Although so, the centroids captured by EulerK deviate from the unit hyper-sphere surface and thus in strict distributional sense, actually are outliers. This weird phenomenon also occurs in some generic kernel clustering methods. Intuitively, using such outlier-like centroids should not be quite reasonable but it is still seldom attended. To eliminate the deviation, we propose two Rectified Euler k-means methods, i.e., REK1 and REK2, which retain the merits of EulerK while acquire real centroids residing on the mapped space to better characterize the data structures. Specifically, REK1 rectifies EulerK by imposing the constraint on the centroids while REK2 views each centroid as the mapped image from a pre-image in the original space and optimizes these pre-images in Euler kernel induced space. Undoubtedly, our proposed REKs can methodologically be extended to solve problems of such a category. Finally, the experiments validate the effectiveness of REK1 and REK2.