  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.
