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

Spectrally approximating large graphs with smaller graphs

How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue distributions, as well as on the ratio between the coarsened and actual graph sizes. Our results carry implications for learning methods that utilize coarsening. For the particular case of spectral clustering, they imply that coarse eigenvectors can be used to derive good quality assignments even without refinement---this phenomenon was previously observed, but lacked formal justification.Comment: 22 pages, 10 figure

A Method Based on Total Variation for Network Modularity Optimization using the MBO Scheme

The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups of nodes called "communities". One popular approach to find communities is to maximize a quality function known as {\em modularity} to achieve some sort of optimal clustering of nodes. In this paper, we interpret the modularity function from a novel perspective: we reformulate modularity optimization as a minimization problem of an energy functional that consists of a total variation term and an $\ell_2$ balance term. By employing numerical techniques from image processing and $\ell_1$ compressive sensing -- such as convex splitting and the Merriman-Bence-Osher (MBO) scheme -- we develop a variational algorithm for the minimization problem. We present our computational results using both synthetic benchmark networks and real data.Comment: 23 page

Topics in High-Dimensional Statistics and the Analysis of Large Hyperspectral Images.

Advancement in imaging technology has made hyperspectral images gathered from remote sensing much more common. The high-dimensional nature of these large scale data coupled with wavelength and spatial dependency necessitates high-dimensional and efficient computation methods to address these issues while producing results that are concise and easy to understand. The thesis addresses these issues by examining high-dimensional methods in the context of hyperspectral image classification, unmixing and wavelength correlation estimation. Chapter 2 re-examines the sparse Bayesian learning (SBL) of linear models in a high-dimensional setting with sparse signal. The hard-thresholded version of the SBL estimator, under orthogonal design, achieves non-asymptotic error rate that is comparable to LASSO. We also establish in the chapter that with high-probability the estimator recovers the sparsity structure of the signal. The ability to recover sparsity structures in high dimensional settings is crucial for unmixing with high-dimensional libraries in the next chapter. In Chapter 3, the thesis investigates the application of SBL on the task of linear/bilinear unmixing and classification of hyperspectral images. The proposed model in this chapter uses latent Markov random fields to classify pixels and account for the spatial dependence between pixels. In the proposed model, the pixels belonging to the same group share the same mixture of pure endmembers. The task of unmixing and classification are performed simultaneously, but this method does not address wavelength dependence. Chapter 4 is a natural extension of the previous chapter that contains the framework to account for both spatial and wavelength dependence in the unmixing of hyperspectral images. The classification of the images are performed using approximate spectral clustering while the unmixing task is performed in tandem with sparse wavelength concentration matrix estimation.PHDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/135893/1/chye_1.pd

Dynamical Systems on Networks: A Tutorial

We give a tutorial for the study of dynamical systems on networks. We focus especially on "simple" situations that are tractable analytically, because they can be very insightful and provide useful springboards for the study of more complicated scenarios. We briefly motivate why examining dynamical systems on networks is interesting and important, and we then give several fascinating examples and discuss some theoretical results. We also briefly discuss dynamical systems on dynamical (i.e., time-dependent) networks, overview software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than original version, some reorganization and also more pointers to interesting direction