Operator Norm Convergence of Spectral Clustering on Level Sets

Following Hartigan [1975], a cluster is defined as a connected component of the t-level set of the underlying density, i.e., the set of points for which the density is greater than t. A clustering algorithm which combines a density estimate with spectral clustering techniques is proposed. Our algorithm is composed of two steps. First, a nonparametric density estimate is used to extract the data points for which the estimated density takes a value greater than t. Next, the extracted points are clustered based on the eigenvectors of a graph Laplacian matrix. Under mild assumptions, we prove the almost sure convergence in operator norm of the empirical graph Laplacian operator associated with the algorithm. Furthermore, we give the typical behavior of the representation of the dataset into the feature space, which establishes the strong consistency of our proposed algorithm.