🤖 AI Summary
Multi-partite network spectral embeddings reside in high-dimensional spaces, yet their intrinsic node representations lie within group-specific low-dimensional subspaces—a geometric structure previously uncharacterized.
Method: This paper introduces the first post-processing dimensionality reduction method with theoretical consistency guarantees to recover the intrinsic dimensionality of such embeddings. Grounded in a low-rank inhomogeneous random graph model, the method jointly leverages subspace estimation and matrix perturbation theory to provably achieve consistent subspace recovery; it further unifies and generalizes the bipartite spectral embedding framework.
Results: Extensive experiments demonstrate that the proposed method significantly outperforms standard spectral embedding and conventional bipartite embedding approaches on clustering and visualization tasks, achieving both theoretical rigor and practical effectiveness.
📝 Abstract
Spectral embedding finds vector representations of the nodes of a network, based on the eigenvectors of its adjacency or Laplacian matrix, and has found applications throughout the sciences. Many such networks are multipartite, meaning their nodes can be divided into groups and nodes of the same group are never connected. When the network is multipartite, this paper demonstrates that the node representations obtained via spectral embedding live near group-specific low-dimensional subspaces of a higher-dimensional ambient space. For this reason we propose a follow-on step after spectral embedding, to recover node representations in their intrinsic rather than ambient dimension, proving uniform consistency under a low-rank, inhomogeneous random graph model. Our method nat-urally generalizes bipartite spectral embedding, in which node representations are obtained by singular value decomposition of the biadjacency or bi-Laplacian matrix. Graph embedding describes a family of tools for representing the nodes of a graph (or network) as points in space. Applications include exploratory analyses such as clustering [1, 2] or visualization [3], and predictive tasks such as classification [4] or The purpose of this article is to develop bespoke statistical methodology for the case when the graph is multipartite. treatment
