  Spectral clustering is one of the most popular clustering approaches with the
capability to handle some challenging clustering problems. Most spectral
clustering methods provide a nonlinear map from the data manifold to a
subspace. Only a little work focuses on the explicit linear map which can be
viewed as the unsupervised distance metric learning. In practice, the selection
of the affinity matrix exhibits a tremendous impact on the unsupervised
learning. While much success of affinity learning has been achieved in recent
years, some issues such as noise reduction remain to be addressed. In this
paper, we propose a novel method, dubbed Adaptive Affinity Matrix (AdaAM), to
learn an adaptive affinity matrix and derive a distance metric from the
affinity. We assume the affinity matrix to be positive semidefinite with
ability to quantify the pairwise dissimilarity. Our method is based on posing
the optimization of objective function as a spectral decomposition problem. We
yield the affinity from both the original data distribution and the widely-used
heat kernel. The provided matrix can be regarded as the optimal representation
of pairwise relationship on the manifold. Extensive experiments on a number of
real-world data sets show the effectiveness and efficiency of AdaAM.
