Graph Learning With Riemannian Optimization for Multi-View Integrative Clustering

Real-world multi-view data may manifest as point-clouds, but their meaningful structure often resides on a lower dimensional manifold embedded in the higher dimensional space. Consequently, existing graph based multi-view algorithms focus primarily on extraction of the low-rank subspaces for clustering. However, simultaneous optimization of the individual graph structures, their contributions/weights along with the clustering subspaces is likely to give a more comprehensive idea of the clusters present in the data set. In this regard, the paper proposes a Riemannian manifold optimization algorithm that harnesses the geometry and structure preserving properties of symmetric positive definite (SPD) manifold and Grassmann manifold for efficient multi-view clustering. The SPD manifold is used to optimize the graph Laplacians corresponding to the individual views, while preserving their symmetricity, positive definiteness, and related properties. The Grassmann manifold, on the other hand, is used to minimize the disagreement between the joint and individual clustering subspaces. Optimization over the Grassmann manifold additionally enforces the clustering solutions to be basis invariant such that all cluster indicator matrices whose columns span the same subspace map to the same clustering solution. A gradient based line-search algorithm, which alternates between two different manifolds, is proposed to optimize the multi-view graph Laplacians and its associated subspaces. The matrix perturbation theory is used to theoretically bound the disagreement between the clustering subspaces. Extensive experiments on several multi-view benchmark and multi-omics cancer data sets establish the effectiveness of the proposed method. The proposed work demonstrates how incorporation of Riemannanian optimization into the graph learning framework can boost multi-view clustering performance.