  Co-clustering targets on grouping the samples (e.g., documents, users) and
the features (e.g., words, ratings) simultaneously. It employs the dual
relation and the bilateral information between the samples and features. In
many realworld applications, data usually reside on a submanifold of the
ambient Euclidean space, but it is nontrivial to estimate the intrinsic
manifold of the data space in a principled way. In this study, we focus on
improving the co-clustering performance via manifold ensemble learning, which
is able to maximally approximate the intrinsic manifolds of both the sample and
feature spaces. To achieve this, we develop a novel co-clustering algorithm
called Relational Multi-manifold Co-clustering (RMC) based on symmetric
nonnegative matrix tri-factorization, which decomposes the relational data
matrix into three submatrices. This method considers the intertype relationship
revealed by the relational data matrix, and also the intra-type information
reflected by the affinity matrices encoded on the sample and feature data
distributions. Specifically, we assume the intrinsic manifold of the sample or
feature space lies in a convex hull of some pre-defined candidate manifolds. We
want to learn a convex combination of them to maximally approach the desired
intrinsic manifold. To optimize the objective function, the multiplicative
rules are utilized to update the submatrices alternatively. Besides, both the
entropic mirror descent algorithm and the coordinate descent algorithm are
exploited to learn the manifold coefficient vector. Extensive experiments on
documents, images and gene expression data sets have demonstrated the
superiority of the proposed algorithm compared to other well-established
methods.
