Efficient Clustering on Riemannian Manifolds: A Kernelised Random Projection Approach

TL;DR

This paper introduces an efficient kernelised random projection framework for clustering on Riemannian manifolds, significantly reducing computational costs while maintaining high clustering performance in computer vision tasks.

Contribution

It proposes a novel framework using kernel space random projections for manifold data, enabling scalable clustering with preserved geometric structure.

Findings

Reduces computational complexity by over two orders of magnitude.

Maintains clustering performance comparable to existing methods.

Validated on multiple computer vision applications.

Abstract

Reformulating computer vision problems over Riemannian manifolds has demonstrated superior performance in various computer vision applications. This is because visual data often forms a special structure lying on a lower dimensional space embedded in a higher dimensional space. However, since these manifolds belong to non-Euclidean topological spaces, exploiting their structures is computationally expensive, especially when one considers the clustering analysis of massive amounts of data. To this end, we propose an efficient framework to address the clustering problem on Riemannian manifolds. This framework implements random projections for manifold points via kernel space, which can preserve the geometric structure of the original space, but is computationally efficient. Here, we introduce three methods that follow our framework. We then validate our framework on several computer vision applications by comparing against popular clustering methods on Riemannian manifolds. Experimental results demonstrate that our framework maintains the performance of the clustering whilst massively reducing computational complexity by over two orders of magnitude in some cases.