Kernel-Based Distance Metric Learning for Supervised $k$ -Means Clustering

Finding an appropriate distance metric that accurately reflects the (dis)similarity between examples is a key to the success of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-means clustering. While it is not always an easy task to specify a good distance metric, we can try to learn one based on prior knowledge from some available clustered data sets, an approach that is referred to as supervised clustering. In this paper, a kernel-based distance metric learning method is developed to improve the practical use of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-means clustering. Given the corresponding optimization problem, we derive a meaningful Lagrange dual formulation and introduce an efficient algorithm in order to reduce the training complexity. Our formulation is simple to implement, allowing a large-scale distance metric learning problem to be solved in a computationally tractable way. Experimental results show that the proposed method yields more robust and better performances on synthetic as well as real-world data sets compared to other state-of-the-art distance metric learning methods.