Learning a consensus affinity matrix for multi-view clustering via subspaces merging on Grassmann manifold
Introduction
Clustering aims to partition data into different groups such that data in the same groups are similar [1], [2], [3]. In such a task, similarity measurement among samples, denoted by an affinity matrix, obtained by various metrics, plays a significant role in affecting the clustering performance. For example, heat kernel-based similarity measurements are widely used in spectral clustering. The performance relies on the heuristic metric and is likely to be corrupted by noise or outliers. To address this drawback, subspace clustering provides a simple yet effective way to find a latent affinity matrix [4].
Subspace clustering uses the self-expressiveness property of data samples. It assumes that each sample can be represented as a linear combination of other samples. Under this assumption, a representation matrix with different regularization can be constructed from samples and is then used to construct the affinity matrix. Low-rank representation (LRR) subspace clustering [5], [6] seeks the lowest rank representation among all the candidates. The low-rank regularization implies that the original data space is spanned by a small number of vectors, which describes the global property of the data subspaces. Sparse subspace clustering (SSC) [7] takes the data matrix as the dictionary and performs the representation matrix calculation as a sparse coding task. The sparse regularization implies that the data sample can be represented by a few other samples. To take advantage of the LRR subspace clustering and SSC, subspace clustering methods considering both low-rank and sparse regularizations [8], [9] are developed to capture the global and local structures of the data space.
Although these subspace clustering approaches are effective, the information provided by single-source data is limited, especially when the observations are insufficient and/or grossly corrupted. In real-world applications, datasets are often with multiple modalities or composed of multiple representations (i.e., views). Multi-view clustering problems aim to partition data into different groups by making use of complementary information from these heterogeneous views [10]. Multi-view learning methods learn a latent representation of data from multiple views assuming that data in all views share a common structure [11], [12]. Feature concatenation is a simple way to combine all data views, however, this method cannot discover common structure across different views. To tackle such a problem, co-regularized multi-view spectral clustering minimizes the eigenvectors of the graph Laplacian across different views to learn a common representation of the multi-view data [13]. Most previous works tend to construct affinity matrices on each individual view before learning the consensus affinity matrix [14], [3]. These methods divide the learning problem into two independent processes without adaptive interaction. In contrast, methods such as multi-view low-rank sparse subspace clustering (MLRSSC) construct the multi-view affinity matrix in one step by learning a common representation matrix from the subspace representations of each view [15]. In addition, latent multi-view subspace clustering (LMSC) assumes that views originate from one underlying latent representation, and then subspace clustering is performed on the latent representation [16]. To deal with incomplete data, multi-view co-clustering with incomplete data [17] finds a consistent cluster partition based on rank-one matrix approximation. All of these methods focus on regularizations on individual representations and consistency preservation between the common representation and individual representations.
However, data from different views usually have different structures. Therefore, direct affinity fusion among different views in Euclidean space is too rigid to align the learned subspaces. Such approaches easily break the local structure of each individual subspace and yield poor performance. To address this issue, this paper proposes to align the subspaces on a Grassmann manifold [18], [19] to learn the consensus affinity matrix. To preserve the local structure within each subspace, the consensus affinity matrix is also regularized to have the same structure properties as those of individual views, which helps learn the common structures instead of mixing or breaking them. The convex optimization problem is solved using the alternating direction method of multipliers (ADMM) [20]. We conduct extensive experiments on synthetic and real-world datasets to demonstrate the performance of the proposed method. Specifically, the main contributions of this paper can be summarized as follows:
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We merge the self-representative subspaces of individual views on a Grassmann manifold to obtain a robust integrative subspace. The obtained integrative subspace preserves the geometric uniformity of the subspaces from each view.
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The affinity matrix is directly learned on the integrative subspace and is further regularized with low-rank and sparse constraints. The regularization over view-specific space and Grassmann manifold ensures the favorable subsequent clustering performance.
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We conduct extensive experiments on synthetic and real-world datasets and demonstrate that our method outperforms several state-of-the-art multi-view clustering methods. The superior performance of the proposed method validates its effectiveness.
The remainder of this paper is as follows. Section 2 briefly reviews the background and related works for multi-view subspace clustering. Section 3 introduces the proposed model and provides the optimization algorithms. Section 4 demonstrates the experimental results on one synthetic dataset and eight real-world datasets. Section 5 presents the conclusion of this paper.
Section snippets
Notation
Throughout the paper, scalars are represented with lower-case symbols, vectors with bold lower-case symbols and matrices with bold upper-case symbols. The i-th column of a matrix is denoted by , whereas denotes its j-th row. The widely used norm, Frobenius norm, norm, and nuclear norm are denoted by and , respectively. The Schatten p-norm is denoted by and defined as where is the singular value decomposition of matrix . We use
Integrative affinity learning for multi-view subspace clustering through grassmann alignment
In this section, we present an integrative learning model for multi-view subspace clustering. There are two major merits that distinguish this approach from other popular models. The first one is that the learned subspace from each individual view is aligned to an integrative subspace on Grassmann manifolds to ensure geometric uniformity. The second one is that a latent integrative affinity matrix is directly estimated, and thus facilitates and enhances the clustering performance. The two terms
Experiments
In this section, we conduct experiments on synthetic data and real-world datasets to evaluate the clustering performance of our method compared with several state-of-the-art algorithms. We also propose an efficient parameter tuning strategy based on the parameter sensitivity.
Conclusion
In this paper, we propose an integrative affinity learning model for multi-view subspace clustering to enhance clustering performance. The method directly learns a consensus affinity matrix by merging subspace representations of multiple individual views on a Grassmann manifold, rather than concentrating on subspace learning or alignment on Euclidean space. We also penalize the consensus affinity matrix to extract the latent common information among multiple views. In addition, we provide a
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This work was partially supported by the Key-Area Research and Development of Guangdong Province (2020B010166002, 2020B111119001), National Natural Science Foundation of China (61771007), Science and Technology Planning Project of Guangdong Province (2017B020226004), and the Health & Medical Collaborative Innovation Project of Guangzhou City (202002020049).
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