Clustering Tree-structured Data on Manifold

TL;DR

This paper introduces a novel framework for clustering tree-structured data on manifolds using a new parameterization called T-A matrix, enabling effective analysis of topological and geometrical information.

Contribution

It proposes the T-A matrix for data parameterization on manifolds and integrates structure constraints into matrix factorization for clustering tree-structured data.

Findings

Effective clustering on simulated data

Accurate clustering on retinal images

Outperforms existing methods

Abstract

Tree-structured data usually contain both topological and geometrical information, and are necessarily considered on manifold instead of Euclidean space for appropriate data parameterization and analysis. In this study, we propose a novel tree-structured data parameterization, called Topology-Attribute matrix (T-A matrix), so the data clustering task can be conducted on matrix manifold. We incorporate the structure constraints embedded in data into the negative matrix factorization method to determine meta-trees from the T-A matrix, and the signature vector of each single tree can then be extracted by meta-tree decomposition. The meta-tree space turns out to be a cone space, in which we explore the distance metric and implement the clustering algorithm based on the concepts like Fr\'echet mean. Finally, the T-A matrix based clustering (TAMBAC) framework is evaluated and compared using both simulated data and real retinal images to illustrate its efficiency and accuracy.