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.
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.
@cite_32
2012619263 2051666324 2885633361 1527474722
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. Previous efforts in hashing intend to preserve data variance or pairwise affinity, but neither is adequate in capturing the manifold structures hidden in most visual data. In this paper, we tackle this problem by reconstructing the locally linear structures of manifolds in the binary Hamming space, which can be learned by locality-sensitive sparse coding. We cast the problem as a joint minimization of reconstruction error and quantization loss, and show that, despite its NP-hardness, a local optimum can be obtained efficiently via alternative optimization. Our method distinguishes itself from existing methods in its remarkable ability to extract the nearest neighbors of the query from the same manifold, instead of from the ambient space. On extensive experiments on various image benchmarks, our results improve previous state-of-the-art by 28-74 typically, and 627 on the Yale face data. Aggregating different image features for image retrieval has recently shown its effectiveness. While highly effective, though, the question of how to uplift the impact of the best features for a specific query image persists as an open computer vision problem. In this paper, we propose a computationally efficient approach to fuse several hand-crafted and deep features, based on the probabilistic distribution of a given membership score of a constrained cluster in an unsupervised manner. First, we introduce an incremental nearest neighbor (NN) selection method, whereby we dynamically select k-NN to the query. We then build several graphs from the obtained NN sets and employ constrained dominant sets (CDS) on each graph G to assign edge weights which consider the intrinsic manifold structure of the graph, and detect false matches to the query. Finally, we elaborate the computation of feature positive-impact weight (PIW) based on the dispersive degree of the characteristics vector. To this end, we exploit the entropy of a cluster membership-score distribution. In addition, the final NN set bypasses a heuristic voting scheme. Experiments on several retrieval benchmark datasets show that our method can improve the state-of-the-art result. Many motion segmentation algorithms based on manifold clustering rely on a accurate rank estimation of the trajectory matrix and on a meaningful affinity measure between the estimated manifolds. While it is known that rank estimation is a difficult task, we also point out the problems that can be induced by an affinity measure that neglects the distribution of the principal angles. In this paper we suggest a new interpretation of the rank of the trajectory matrix and a new affinity measure. The rank estimation is performed by analysing which rank leads to a configuration where small and large angles are best separated. The affinity measure is a new function automatically parametrized so that it is able to adapt to the actual configuration of the principal angles. Our technique has one of lowest misclassification rates on the Hopkins155 database and has good performances also on synthetic sequences with up to 5 motions and variable noise level.