Convergence of the Graph Allen–Cahn Scheme

Networks, which represent agents and interactions between them, arise in myriad applications throughout the sciences, engineering, and even the humanities. To understand large-scale structure in a network, a common task is to cluster a network's nodes into sets called "communities", such that there are dense connections within communities but sparse connections between them. A popular and statistically principled method to perform such clustering is to use a family of generative models known as stochastic block models (SBMs). In this paper, we show that maximum likelihood estimation in an SBM is a network analog of a well-known continuum surface-tension problem that arises from an application in metallurgy.To illustrate the utility of this relationship, we implement network analogs of three surface-tension algorithms, with which we successfully recover planted community structure in synthetic networks and which yield fascinating insights on empirical networks that we construct from hyperspectral videos.(MBO) scheme, geometric partial differential equations AMS subject classifications. 65K10, 49M20, 35Q56, 62H30, 91C20, 91D30, 94C151. Introduction. The study of networks, in which nodes represent entities and edges encode interactions between entities [66], can provide useful insights into a wide variety of complex systems in myriad fields, such as granular materials [73], disease spreading [74], criminology [37], and more. In the study of such applications, the analysis of large data sets -from diverse sources and applications -continues to grow ever more important.The simplest type of network is a graph, and empirical networks often appear to exhibit a complicated mixture of regular and seemingly random features [66]. Additionally, it is increasingly important to study networks with more complicated features, such as time-dependence [39], multiplexity [50], annotations [67], and connections that go beyond a pairwise paradigm [72]. One also has to worry about "features" such as missing information and false positives [48]. Nevertheless, it is convenient in the present paper to restrict our attention to undirected, unweighted graphs for simplicity.To try to understand the large-scale structure of a network, it can be very insightful to coarse-grain it in various ways [27,79,81,83,84]. The most popular type of clustering is the detection of assortative "communities," in which dense sets of nodes are connected sparsely to other dense sets of nodes [27,81]. A statistically principled approach is to treat community detection as a statistical inference problem using a model such * Submitted to the editors DATE.

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“…where c > 2/ [54]. Using the constant c leads to an unconditionally stable scheme, which negates the stiffness caused by the 1/ scale.…”

Section: (Upper Bound)mentioning

confidence: 99%

Stochastic Block Models are a Discrete Surface Tension

Boyd

Porter²,

Bertozzi³

2019

J Nonlinear Sci

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“…(see [42,23] for details). Here ψ(u) = (u 2 − 1) 2 is the double-well potential, which we understand to be applied component-wise, and Ω denotes a diagonal matrix with entries Ω ii = ω 0 > 0 if vertex i belongs to the training data and Ω ii = 0 otherwise.…”

Section: Spectral Clusteringmentioning

confidence: 99%

NFFT Meets Krylov Methods: Fast Matrix-Vector Products for the Graph Laplacian of Fully Connected Networks

Alfke

Potts

Stoll

et al. 2018

Front. Appl. Math. Stat.

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The graph Laplacian is a standard tool in data science, machine learning, and image processing. The corresponding matrix inherits the complex structure of the underlying network and is in certain applications densely populated. This makes computations, in particular matrix-vector products, with the graph Laplacian a hard task. A typical application is the computation of a number of its eigenvalues and eigenvectors. Standard methods become infeasible as the number of nodes in the graph is too large. We propose the use of the fast summation based on the nonequispaced fast Fourier transform (NFFT) to perform the dense matrix-vector product with the graph Laplacian fast without ever forming the whole matrix. The enormous flexibility of the NFFT algorithm allows us to embed the accelerated multiplication into Lanczos-based eigenvalues routines or iterative linear system solvers and even consider other than the standard Gaussian kernels. We illustrate the feasibility of our approach on a number of test problems from image segmentation to semi-supervised learning based on graph-based PDEs. In particular, we compare our approach with the Nyström method. Moreover, we present and test an enhanced, hybrid version of the Nyström method, which internally uses the NFFT.

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“…This is the Allen-Cahn Equation [1,30] with fidelity term with the differential operator ∆u replaced by a more general graph operator −L s [51]; when → 0, the solution to the Allen-Cahn equation approximates motion by mean curvature [62]. Note that in the last term of (7), the product is meant to be calculated on each node.…”

Section: Semi-supervised Algorithmmentioning

confidence: 99%

Hyperspectral Image Classification Using Graph Clustering Methods

Zhang¹,

Merkurjev²,

Koniges³

et al. 2017

Image Processing on Line

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Hyperspectral imagery is a challenging modality due to the dimension of the pixels which can range from hundreds to over a thousand frequencies depending on the sensor. Most methods in the literature reduce the dimension of the data using a method such as principal component analysis, however this procedure can lose information. More recently methods have been developed to address classification of large datasets in high dimensions. This paper presents two classes of graph-based classification methods for hyperspectral imagery. Using the full dimensionality of the data, we consider a similarity graph based on pairwise comparisons of pixels. The graph is segmented using a pseudospectral algorithm for graph clustering that requires information about the eigenfunctions of the graph Laplacian but does not require computation of the full graph. We develop a parallel version of the Nyström extension method to randomly sample the graph to construct a low rank approximation of the graph Laplacian. With at most a few hundred eigenfunctions, we can implement the clustering method designed to solve a variational problem for a graph-cut-based semi-supervised or unsupervised classification problem. We implement OpenMP directive-based parallelism in our algorithms and show performance improvement and strong, almost ideal, scaling behavior. The method can handle very large datasets including a video sequence with over a million pixels, and the problem of segmenting a data set into a pre-determined number of classes. Source CodeThe reviewed source code and documentation for this algorithm are available from the web page of this article 1 . Compilation and usage instructions are included in the README.txt file of the archive.

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Convergence of the Graph Allen–Cahn Scheme

Cited by 28 publications

References 31 publications

Stochastic Block Models are a Discrete Surface Tension

Stochastic Block Models are a Discrete Surface Tension

NFFT Meets Krylov Methods: Fast Matrix-Vector Products for the Graph Laplacian of Fully Connected Networks

Hyperspectral Image Classification Using Graph Clustering Methods

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