Multiple Kernel $k$-means Clustering using Min-Max Optimization with $l_2$ Regularization

As various types of biomedical data become available, multiple kernel learning approaches have been proposed to incorporate abundant yet diverse information collected from multiple sources (or views) to facilitate disease prediction and pattern recognition. Although supervised multiple kernel learning has been extensively studied, until recently, only a few unsupervised approaches have been proposed. Moreover, the existing unsupervised approaches are unable to effectively utilize useful and complementary information especially when signals in some views are weak. We propose a novel multiple kernel $k$-means clustering method which aims to effectively use complementary information from multiple views to identify clusters. It is achieved by optimizing the unsupervised problem using a $\min_{\mathbf{H}}$-$\max_{\mathbf{\theta}}$ formulation, such that more weights can be assigned to views having weak signal for cluster identification. Moreover, our method avoids dismissing views with informative but weak signals by imposing $l_2$ constraint. Additionally, it allows to distill biological prior knowledge on the clustering by imposing a linear constraint on the kernel coefficients. To evaluate our method, we compare it with seven other clustering approaches on simulated multiview data. The simulation results show that our method outperforms existing clustering approaches especially when there is noise and redundancy in the data.