Rectified Euler k-means and beyond. (May 2023)

• Record Type: Journal Article
• Authors: Lin, Yunxia; Chen, Songcan
• Is Part Of: Pattern recognition. Volume 137(2023)
• Publication Date: 2023-05
• Subjects: Kernel k-means -- Euler kernel -- Pseudo centroid -- Rectified euler k-means; 00-01 -- 99-00; Pattern perception -- Periodicals; Perception des structures -- Périodiques; Patroonherkenning; 006.4
• DOI: 10.1016/j.patcog.2022.109283
• Languages: English
• ISSNs: 0031-3203

Abstract

Highlights: We first find that EulerK actually acquires the outlier-like centroids. We introduce two rectified versions of EulerK named REK1 and REK2 to rectify this oddity. Our proposed REK1 and REK2 can methodologically be extended straightforwardly to deal with problems of such a category beyond Euler k -means. The experimental results on a synthetic dataset and several commonly used real datasets validate the rationality and effectiveness of our proposed REK1 and REK2. We also find an interesting phenomenon about the parameter α . 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 R ectified E uler k -means methods, i.e., REK1 and REK2, which retain the merits of EulerK while acquiring 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.