14 research outputs found
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