Coclustering targets on grouping the samples (e.g., documents and users) and the features (e.g., words and ratings) simultaneously. It employs the dual relation and the bilateral information between the samples and features. In many real-world 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 paper, we focus on improving the coclustering 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 coclustering algorithm called relational multimanifold coclustering based on symmetric nonnegative matrix trifactorization, which decomposes the relational data matrix into three submatrices. This method considers the intertype relationship revealed by the relational data matrix and also the intratype information reflected by the affinity matrices encoded on the sample and feature data distributions. Specifically, we assume that the intrinsic manifold of the sample or feature space lies in a convex hull of some predefined 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. In addition, 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 with other well-established methods.
@cite_32: We propose an automatic approximation of the intrinsic manifold for general semi-supervised learning problems. Unfortunately, it is not trivial to define an optimization function to obtain optimal hyperparameters. Usually, pure cross-validation is considered but it does not necessarily scale up. A second problem derives from the suboptimality incurred by discrete grid search and overfitting problems. As a consequence, we developed an ensemble manifold regularization (EMR) framework to approximate the intrinsic manifold by combining several initial guesses. Algorithmically, we designed EMR very carefully so that it (a) learns both the composite manifold and the semi-supervised classifier jointly; (b) is fully automatic for learning the intrinsic manifold hyperparameters implicitly; (c) is conditionally optimal for intrinsic manifold approximation under a mild and reasonable assumption; and (d) is scalable for a large number of candidate manifold hyperparameters, from both time and space perspectives. Extensive experiments over both synthetic and real datasets show the effectiveness of the proposed framework.
However, these graph regularized methods have a common shortcoming that the manifold used for co-clustering might not be the true intrinsic manifold, and it will even deviates far from the desired in an adverse situation. It is a nontrivial task to seek the intrinsic manifold in reality. To alleviate this difficulty, in light of @cite_32 , we assume the intrinsic manifold of the sample or feature distribution lies in a convex hull of some candidate manifolds, and hope to maximally approximate the true intrinsic manifold using the convex combination of them. This way, the local geometrical structure can be better preserved in both the sample and feature spaces for co-clustering.