Spectral clustering with eigenvector selection

The task of discovering natural groupings of input patterns, or clustering, is an important aspect machine learning and pattern analysis. In this paper, we study the widely-used spectral clustering algorithm which clusters data using eigenvectors of a similarity/affinity matrix derived from a data set. In particular, we aim to solve two critical issues in spectral clustering: (1) How to automatically determine the number of clusters? and (2) How to perform effective clustering given noisy and sparse data? An analysis of the characteristics of eigenspace is carried out which shows that (a) Not every eigenvectors of a data affinity matrix is informative and relevant for clustering; (b) Eigenvector selection is critical because using uninformative/irrelevant eigenvectors could lead to poor clustering results; and (c) The corresponding eigenvalues cannot be used for relevant eigenvector selection given a realistic data set. Motivated by the analysis, a novel spectral clustering algorithm is proposed which differs from previous approaches in that only informative/relevant eigenvectors are employed for determining the number of clusters and performing clustering. The key element of the proposed algorithm is a simple but effective relevance learning method which measures the relevance of an eigenvector according to how well it can separate the data set into different clusters. Our algorithm was evaluated using synthetic data sets as well as real-world data sets generated from two challenging visual learning problems. The results demonstrated that our algorithm is able to estimate the cluster number correctly and reveal natural grouping of the input data/patterns even given sparse and noisy data.