Enhancing Kernel Power $K$-means: Scalable and Robust Clustering with Random Fourier Features and Possibilistic Method

Abstract

Kernel power k-means (KPKM) leverages a family of means to mitigate local minima issues in kernel k-means. However, KPKM faces two key limitations: (1) the computational burden of the full kernel matrix restricts its use on extensive data, and (2) the lack of authentic centroid-sample assignment learning reduces its noise robustness. To overcome these challenges, we propose RFF-KPKM, introducing the first approximation theory for applying random Fourier features (RFF) to KPKM. RFF-KPKM employs RFF to generate efficient, low dimensional feature maps, bypassing the need for the whole kernel matrix. Crucially, we are the first to establish strong theoretical guarantees for this combination: (1) an excess risk bound of O( k^3/n), (2) strong consistency with membership values, and (3) a (1 + ε) relative error bound achievable using the RFF of dimension poly(ε^{−1} logk). Furthermore, to improve robustness and the ability to learn multiple kernels, we propose IP-RFF-MKPKM, an improved possibilistic RFF-based multiple kernel power k-means. IP-RFF-MKPKM ensures the scalability of MKPKM via RFF and refines cluster assignments by combining the merits of the possibilistic and fuzzy membership. Experiments on large-scale datasets demonstrate the superior efficiency and clustering accuracy of the proposed methods compared to the state-of-the-art alternatives.