Laplacian mixture modeling for network analysis and unsupervised learning on graphs

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.

TopOMetry systematically learns and evaluates the latent dimensions of single-cell atlases

“…A particularity of methods grounded on discrete differential geometry such as LE, DM, and UMAP is their ability to independently achieve discrete approximations of the Laplace-Beltrami Operator (LBO) 19,22,25,41 , a generalization of the graph Laplacian and an optimal tool to recover information about data intrinsic structure 19,42 . Methods that approximate the LBO are advantageous due to their natural connection to spectral theory and clustering and are intuitively related to the graph smoothing process employed by these methods 19,31,32,41,43 , such as in continuous k-nearest-neighbors (CkNN), which can also approximate higher-order operators (the 0-th Laplace-de Rham Operator is the Laplace-Beltrami Operator) 32 .…”

“…A particularity of methods grounded on discrete differential geometry such as LE, DM, and UMAP is their ability to independently achieve discrete approximations of the Laplace-Beltrami Operator (LBO) 19,22,25,41 , a generalization of the graph Laplacian and an optimal tool to recover information about data intrinsic structure 19,42 . Methods that approximate the LBO are advantageous due to their natural connection to spectral theory and clustering and are intuitively related to the graph smoothing process employed by these methods 19,31,32,41,43 , such as in continuous k-nearest-neighbors (CkNN), which can also approximate higher-order operators (the 0-th Laplace-de Rham Operator is the Laplace-Beltrami Operator) 32 . Some work has been done in comparing the performance of distinct methods for DR in single-cell data [44][45][46] -yet, these are limited to quality metrics in benchmarking pipelines and lack both the needed tuning of each algorithm to the analyzed data sets and the qualitative assessment of each algorithm's topology preservation of actual ground-truth biological landmarks, such as the cell cycle as a closed logic loop.…”