Riemannian Multi-Manifold Modeling

TL;DR

This paper introduces a Riemannian manifold clustering framework that leverages intrinsic geometry, local tangent space information, and spectral clustering, demonstrating effectiveness on various complex datasets.

Contribution

The paper presents a novel Riemannian clustering algorithm that exploits intrinsic geometric features and provides theoretical guarantees even with intersecting clusters.

Findings

Effective clustering on sphere, positive definite matrices, and Grassmannian.

Superior performance over existing methods on synthetic and real data.

Robustness to deviations from theoretical assumptions.

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.