Nonlinear Adaptive Distance Metric Learning for Clustering

A good distance metric is crucial for many data mining tasks. To learn a metric in the unsupervised setting, most metric learning algorithms project observed data to a lowdimensional manifold, where geometric relationships such as pairwise distances are preserved. It can be extended to the nonlinear case by applying the kernel trick, which embeds the data into a feature space by specifying the kernel function that computes the dot products between data points in the feature space. In this paper, we propose a novel unsupervised Nonlinear AdaptiveMetric Learning algorithm, called NAML, which performs clustering and distance metric learning simultaneously. NAML first maps the data to a high-dimensional space through a kernel function; then applies a linear projection to find a low-dimensional manifold where the separability of the data is maximized; and finally performs clustering in the low-dimensional space. The performance of NAML depends on the selection of the kernel function and the projection. We show that the joint kernel learning, dimensionality reduction, and clustering can be formulated as a trace maximization problem, which can be solved via an iterative procedure in the EM framework. Experimental results demonstrated the efficacy of the proposed algorithm.