Integrate and Conquer

In this article, we introduce a novel, general methodology, called integrate and conquer, for simultaneously accomplishing the tasks of feature extraction, manifold construction, and clustering, which is taken to be superior to building a clustering method as a single task. When the proposed novel methodology is used on two-dimensional (2D) data, it naturally induces a new clustering method highly effective on 2D data. Existing clustering algorithms usually need to convert 2D data to vectors in a preprocessing step, which, unfortunately, severely damages 2D spatial information and omits inherent structures and correlations in the original data. The induced new clustering method can overcome the matrix-vectorization-related issues to enhance the clustering performance on 2D matrices. More specifically, the proposed methodology mutually enhances three tasks of finding subspaces, learning manifolds, and constructing data representation in a seamlessly integrated fashion. When used on 2D data, we seek two projection matrices with optimal numbers of directions to project the data into low-rank, noise-mitigated, and the most expressive subspaces, in which manifolds are adaptively updated according to the projections, and new data representation is built with respect to the projected data by accounting for nonlinearity via adaptive manifolds. Consequently, the learned subspaces and manifolds are clean and intrinsic, and the new data representation is discriminative and robust. Extensive experiments have been conducted and the results confirm the effectiveness of the proposed methodology and algorithm.