  Spectral Clustering (SC) is one of the most widely used methods for data
clustering. It first finds a low-dimensonal embedding $U$ of data by computing
the eigenvectors of the normalized Laplacian matrix, and then performs k-means
on $U^\top$ to get the final clustering result. In this work, we observe that,
in the ideal case, $UU^\top$ should be block diagonal and thus sparse.
Therefore we propose the Sparse Spectral Clustering (SSC) method which extends
SC with sparse regularization on $UU^\top$. To address the computational issue
of the nonconvex SSC model, we propose a novel convex relaxation of SSC based
on the convex hull of the fixed rank projection matrices. Then the convex SSC
model can be efficiently solved by the Alternating Direction Method of
\canyi{Multipliers} (ADMM). Furthermore, we propose the Pairwise Sparse
Spectral Clustering (PSSC) which extends SSC to boost the clustering
performance by using the multi-view information of data. Experimental
comparisons with several baselines on real-world datasets testify to the
efficacy of our proposed methods.
