On unsupervised simultaneous kernel learning and data clustering

A novel optimization framework for joint unsupervised clustering and kernel learning is derived. Sparse nonnegative matrix factorization of kernel covariance matrices is utilized to categorize data according to their information content. It is demonstrated that a pertinent kernel covariance matrix for clustering can be constructed such that it is block diagonal within arbitrary row and column permutations, while each diagonal block has rank one. To achieve this, a linear combination of a dictionary of kernels is sought such that it has rank equal to the number of clusters while a certain kernel eigenvalue is maximized by a novel difference of convex functions formulation. We establish that the proposed algorithm converges to a stationary solution. Numerical tests with different datasets demonstrate the effectiveness of the proposed scheme whose performance is very close to supervised methods, and performs better than unsupervised alternatives without the need of painstaking parameter tuning. (C) 2020 Elsevier Ltd. All rights reserved.