24 research outputs found
Hearing the clusters in a graph: A distributed algorithm
We propose a novel distributed algorithm to cluster graphs. The algorithm
recovers the solution obtained from spectral clustering without the need for
expensive eigenvalue/vector computations. We prove that, by propagating waves
through the graph, a local fast Fourier transform yields the local component of
every eigenvector of the Laplacian matrix, thus providing clustering
information. For large graphs, the proposed algorithm is orders of magnitude
faster than random walk based approaches. We prove the equivalence of the
proposed algorithm to spectral clustering and derive convergence rates. We
demonstrate the benefit of using this decentralized clustering algorithm for
community detection in social graphs, accelerating distributed estimation in
sensor networks and efficient computation of distributed multi-agent search
strategies
Spectral clustering and the high-dimensional stochastic blockmodel
Networks or graphs can easily represent a diverse set of data sources that
are characterized by interacting units or actors. Social networks, representing
people who communicate with each other, are one example. Communities or
clusters of highly connected actors form an essential feature in the structure
of several empirical networks. Spectral clustering is a popular and
computationally feasible method to discover these communities. The stochastic
blockmodel [Social Networks 5 (1983) 109--137] is a social network model with
well-defined communities; each node is a member of one community. For a network
generated from the Stochastic Blockmodel, we bound the number of nodes
"misclustered" by spectral clustering. The asymptotic results in this paper are
the first clustering results that allow the number of clusters in the model to
grow with the number of nodes, hence the name high-dimensional. In order to
study spectral clustering under the stochastic blockmodel, we first show that
under the more general latent space model, the eigenvectors of the normalized
graph Laplacian asymptotically converge to the eigenvectors of a "population"
normalized graph Laplacian. Aside from the implication for spectral clustering,
this provides insight into a graph visualization technique. Our method of
studying the eigenvectors of random matrices is original.Comment: Published in at http://dx.doi.org/10.1214/11-AOS887 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Note on Perturbation Results for Learning Empirical Operators
A large number of learning algorithms, for example, spectral clustering, kernel Principal Components Analysis and many manifold methods are based on estimating eigenvalues and eigenfunctions of operators defined by a similarity function or a kernel, given empirical data. Thus for the analysis of algorithms, it is an important problem to be able to assess the quality of such approximations. The contribution of our paper is two-fold: 1. We use a technique based on a concentration inequality for Hilbert spaces to provide new much simplified proofs for a number of results in spectral approximation. 2. Using these methods we provide several new results for estimating spectral properties of the graph Laplacian operator extending and strengthening results from [26]
An interactive analysis of harmonic and diffusion equations on discrete 3D shapes
AbstractRecent results in geometry processing have shown that shape segmentation, comparison, and analysis can be successfully addressed through the spectral properties of the Laplace–Beltrami operator, which is involved in the harmonic equation, the Laplacian eigenproblem, the heat diffusion equation, and the definition of spectral distances, such as the bi-harmonic, commute time, and diffusion distances. In this paper, we study the discretization and the main properties of the solutions to these equations on 3D surfaces and their applications to shape analysis. Among the main factors that influence their computation, as well as the corresponding distances, we focus our attention on the choice of different Laplacian matrices, initial boundary conditions, and input shapes. These degrees of freedom motivate our choice to address this study through the executable paper, which allows the user to perform a large set of experiments and select his/her own parameters. Finally, we represent these distances in a unified way and provide a simple procedure to generate new distances on 3D shapes
Graph Laplacians and their convergence on random neighborhood graphs
Given a sample from a probability measure with support on a submanifold in
Euclidean space one can construct a neighborhood graph which can be seen as an
approximation of the submanifold. The graph Laplacian of such a graph is used
in several machine learning methods like semi-supervised learning,
dimensionality reduction and clustering. In this paper we determine the
pointwise limit of three different graph Laplacians used in the literature as
the sample size increases and the neighborhood size approaches zero. We show
that for a uniform measure on the submanifold all graph Laplacians have the
same limit up to constants. However in the case of a non-uniform measure on the
submanifold only the so called random walk graph Laplacian converges to the
weighted Laplace-Beltrami operator.Comment: Improved presentation, typos corrected, to appear in JML