Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

TL;DR

This paper introduces a scalable Laplacian mixture modeling approach for network analysis and unsupervised learning, enabling probabilistic domain decompositions and low-dimensional representations of graph data.

Contribution

It combines Laplacian eigenspace and mixture modeling to provide a novel, scalable method for clustering and dimensionality reduction on graphs, with provable optimal recovery.

Findings

Provable optimal recovery for cluster graphs

Empirical validation of high-performance heuristics

Connections demonstrated to PageRank and community detection

Abstract

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.