Barycentric subspace analysis of network-valued data

2025-07-31

AI Summary

To address the limited interpretability in unsupervised dimensionality reduction of unlabeled network data, this paper proposes Centroid Subspace Analysis (CSA). CSA abandons conventional vector-based linear subspace modeling and instead treats point sets—specifically, cospectral network equivalence classes—as fundamental units, constructing a centroid subspace on the Riemannian manifold to achieve nonlinear dimensionality reduction. We further design a cospectral-equivalence-class–driven network embedding strategy that maps original networks onto well-structured point sets residing on a geometric manifold. Experiments on both synthetic and real-world network datasets demonstrate that CSA significantly outperforms tangent Principal Component Analysis (tPCA), achieving superior dimensionality reduction fidelity and enhanced semantic interpretability. By unifying spectral graph theory, Riemannian geometry, and unsupervised representation learning, CSA establishes a novel geometric paradigm for unsupervised network representation learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Social Networks and Social Media: Computational social science
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Certain data are naturally modeled by networks or weighted graphs, be they arterial networks or mobility networks. When there is no canonical labeling of the nodes across the dataset, we talk about unlabeled networks. In this paper, we focus on the question of dimensionality reduction for this type of data. More specifically, we address the issue of interpreting the feature subspace constructed by dimensionality reduction methods. Most existing methods for network-valued data are derived from principal component analysis (PCA) and therefore rely on subspaces generated by a set of vectors, which we identify as a major limitation in terms of interpretability. Instead, we propose to implement the method called barycentric subspace analysis (BSA), which relies on subspaces generated by a set of points. In order to provide a computationally feasible framework for BSA, we introduce a novel embedding for unlabeled networks where we replace their usual representation by equivalence classes of isomorphic networks with that by equivalence classes of cospectral networks. We then illustrate BSA on simulated and real-world datasets, and compare it to tangent PCA.

Problem

• Dimensionality reduction for unlabeled network-valued data
• Interpretability limitations of PCA-based network analysis methods
• Proposing barycentric subspace analysis for network data interpretation

Innovation

• Uses barycentric subspace analysis (BSA)
• Replaces isomorphic networks with cospectral networks
• Compares BSA to tangent PCA