Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

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.Comment: 13 figures, 35 reference

Macrostate Data Clustering

We develop an effective nonhierarchical data clustering method using an analogy to the dynamic coarse graining of a stochastic system. Analyzing the eigensystem of an interitem transition matrix identifies fuzzy clusters corresponding to the metastable macroscopic states (macrostates) of a diffusive system. A "minimum uncertainty criterion" determines the linear transformation from eigenvectors to cluster-defining window functions. Eigenspectrum gap and cluster certainty conditions identify the proper number of clusters. The physically motivated fuzzy representation and associated uncertainty analysis distinguishes macrostate clustering from spectral partitioning methods. Macrostate data clustering solves a variety of test cases that challenge other methods.Comment: keywords: cluster analysis, clustering, pattern recognition, spectral graph theory, dynamic eigenvectors, machine learning, macrostates, classificatio

A Lorentz invariance respectful paradigm to explain the origin of ultra-high-energy cosmic ray energies

The issue of this article, whose approach consists in rigorously applying the principle of least action and the invariance of Minkowski space-time, is to explore and extend the equations of special relativity when velocities are greater than c while preserving their covariant nature. This approach, adapted to the study of special relativity, provides a privileged theoretical framework for probing the properties of the superluminal regime if it exists. Relativistic superluminal equations indicate that the speed of light is a singularity in the speeds' spectrum and that the evolution of these equations as a function of the velocity is reversed with respect to the equations of special relativity. This extension of special relativity makes it possible to formulate a credible hypothesis on the origin of ultra-high-energy cosmic ray energies