Riemannian Multi-Manifold Modeling

Abstract

This paper advocates a novel framework for segmenting a dataset in a Riemannian manifold $M$ into clusters lying around low-dimensional submanifolds of $M$. Important examples of $M$, for which the proposed clustering algorithm is computationally efficient, are the sphere, the set of positive definite matrices, and the Grassmannian. The clustering problem with these examples of $M$ is already useful for numerous application domains such as action identification in video sequences, dynamic texture clustering, brain fiber segmentation in medical imaging, and clustering of deformed images. The proposed clustering algorithm constructs a data-affinity matrix by thoroughly exploiting the intrinsic geometry and then applies spectral clustering. The intrinsic local geometry is encoded by local sparse coding and more importantly by directional information of local tangent spaces and geodesics. Theoretical guarantees are established for a simplified variant of the algorithm even when the clusters intersect. To avoid complication, these guarantees assume that the underlying submanifolds are geodesic. Extensive validation on synthetic and real data demonstrates the resiliency of the proposed method against deviations from the theoretical model as well as its superior performance over state-of-the-art techniques.

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• Robust Recovery of Subspace Structures by Low-Rank Representation
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Cited by in corpus (5)

• Deep Unsupervised Clustering Using Mixture of Autoencoders
• Spectral Convergence of the connection Laplacian from random samples
• The Shape of Data and Probability Measures
• Network Clustering Via Kernel-ARMA Modeling and the Grassmannian The Brain-Network Case
• Brain-Network Clustering via Kernel-ARMA Modeling and the Grassmannian