Generalizing Diffuse Interface Methods on Graphs: Nonsmooth Potentials and Hypergraphs

Bosch, Jessica; Klamt, Steffen; Stoll, Martin

doi:10.1137/17m1117835

Cited by 21 publications

(23 citation statements)

References 42 publications

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“…Hypergraphs are encountered in many applications such as biological networks [165], image processing [304], social networks [309], or music recommendation [43]. Hypergraphs have also been used in the context of semi-supervised learning [34,177,227] and particular in convolutional neural networks based on hypergraphs [6,297].…”

Section: The Graph Laplacian Operatormentioning

confidence: 99%

A literature survey of matrix methods for data science

Stoll

2020

GAMM-Mitteilungen

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Efficient numerical linear algebra is a core ingredient in many applications across almost all scientific and industrial disciplines. With this survey we want to illustrate that numerical linear algebra has played and is playing a crucial role in enabling and improving data science computations with many new developments being fueled by the availability of data and computing resources. We highlight the role of various different factorizations and the power of changing the representation of the data as well as discussing topics such as randomized algorithms, functions of matrices, and high‐dimensional problems. We briefly touch upon the role of techniques from numerical linear algebra used within deep learning.

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Section: The Graph Laplacian Operatormentioning

confidence: 99%

A literature survey of matrix methods for data science

Stoll

2020

GAMM-Mitteilungen

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“…Our work has been republished as a SIGEST paper Bertozzi and Flenner [2016]. Over 50 new papers and methods have arises from this work including fast methods for nonlocal means image processing using the MBO scheme Merkurjev, Kostic, and Bertozzi [2013], multiclass learning methods Garcia-Cardona, Merkurjev, Bertozzi, Flenner, and Percus [2014] and Iyer, Chanussot, and Bertozzi [2017], parallel methods for exascale-ready platforms Meng, Koniges, He, S. Williams, Kurth, Cook, Deslippe, and Bertozzi [2016], hyperspectral video analysis Hu, Sunu, and Bertozzi [2015], Merkurjev, Sunu, and Bertozzi [2014], Meng, Merkurjev, Koniges, and Bertozzi [2017], and W. Zhu, Chayes, Tiard, S. Sanchez, Dahlberg, Bertozzi, Osher, Zosso, and Kuang [2017], modularity optimization for network analysis Hu, Laurent, Porter, and Bertozzi [2013] and Boyd, Bai, X. C. Tai, and Bertozzi [2017], measurement techniques in Zoology Calatroni, van Gennip, Schönlieb, Rowland, and Flenner [2017], generalizations to hypergraphs Bosch, Klamt, and Stoll [2016], Pagerank Merkurjev, Bertozzi, and F. Chung [2016] and Cheeger cut based methods Merkurjev, Bertozzi, Yan, and Lerman [2017]. This paper reviews some of this literature and discusses future problem areas including crossover work between network modularity and machine learning and efforts in uncertainty quantification.…”

Section: Data Classification and The Ginzburg-landau Functional On Grmentioning

confidence: 99%

Graphical Models in Machine Learning, Networks and Uncertainty Quantification

Bertozzi

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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This paper is a review article on semi-supervised and unsupervised graph models for classification using similarity graphs and for community detection in networks. The paper reviews graph-based variational models built on graph cut metrics. The equivalence between the graph mincut problem and total variation minimization on the graph for an assignment function allows one to cast graph-cut variational problems in the language of total variation minimization, thus creating a parallel between low dimensional data science problems in Euclidean space (e.g. image segmentation) and high dimensional clustering. The connection paves the way for new algorithms for data science that have a similar structure to well-known computational methods for nonlinear partial differential equations. This paper focuses on a class of methods build around diffuse interface models (e.g. the Ginzburg-Landau functional and the Allen-Cahn equation) and threshold dynamics, developed by the Author and collaborators. Semi-supervised learning with a small amount of training data can be carried out in this framework with diverse applications ranging from hyperspectral pixel classification to identifying activity in police body worn video. It can also be extended to the context of uncertainty quantification with Gaussian noise models. The problem of community detection in networks also has a graph-cut structure and algorithms are presented for the use of threshold dynamics for modularity optimization. With efficient methods, this allows for the use of network modularity for unsupervised machine learning problems with unknown number of classes.

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“…is used in the graph Allen-Cahn equation (as for example in [10]), instead of the smooth potential W introduced above, and if λ := τ /ε = 1, then the graph MBO scheme is equivalent to the following implicit Euler semi-discretisation of the Allen-Cahn equation:…”

Section: Connections Between the Graph Allen-cahn Equation Graph Mbomentioning

confidence: 99%