A semi-supervised heat kernel pagerank MBO algorithm for data classification

Merkurjev, Ekatherina; Bertozzi, Andrea L.; Chung, Fan

doi:10.4310/cms.2018.v16.n5.a4

Cited by 24 publications

(22 citation statements)

References 49 publications

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“…Hence, for all τ ≥ 0 we 10 Since e −τ ∆ χS 1 = 1 2 if and only if e −((n−1) 1−r +1)ω 1−r τ = 0 (which has no solution), and for…”

Section: Star Graphmentioning

confidence: 98%

See 1 more Smart Citation

Graph MBO on Star Graphs and Regular Trees. With Corrections to DOI 10.1007/s00032-014-0216-8

Gennip

2019

Milan J. Math.

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The graph Merriman-Bence-Osher scheme produces, starting from an initial node subset, a sequence of node sets obtained by iteratively applying graph diffusion and thresholding to the characteristic (or indicator) function of the node subsets. One result in [14] gives sufficient conditions on the diffusion time to ensure that the set membership of a given node changes in one iteration of the scheme. In particular, these conditions only depend on local information at the node (information about neighbors and neighbors of neighbors of the node in question). In this paper we show that there does not exist any graph which satisfies these conditions. To make up for this negative result, this paper also presents positive results regarding the Merriman-Bence-Osher dynamics on star graphs and regular trees. In particular, we present sufficient (and in some cases necessary) results for the set membership of a given node to change in one iteration.Mathematics Subject Classification (2010). Primary 35R02, 49K15; Secondary 53C44, 35K05, 05C81.

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“…Hence, for all τ ≥ 0 we 10 Since e −τ ∆ χS 1 = 1 2 if and only if e −((n−1) 1−r +1)ω 1−r τ = 0 (which has no solution), and for…”

Section: Star Graphmentioning

confidence: 98%

“…In recent years the graph version of this process and variations thereof have been succesfully applied to data clustering and classifcation problems and other graph based problems, e.g. in [4,5,8,9,10,7,15,3], which in turn has prompted further theoretical study of the MBO scheme on graphs [14,1].…”

Section: Introductionmentioning

confidence: 99%

Graph MBO on Star Graphs and Regular Trees. With Corrections to DOI 10.1007/s00032-014-0216-8

Gennip

2019

Milan J. Math.

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“…0] converges to a partial optimum [17]. By further simplifying (31) and solving (30) in closed form, we obtain…”

Section: Adaptive Diffusions Robust To Anomaliesmentioning

confidence: 99%

Adaptive Diffusions for Scalable Learning Over Graphs

Berberidis

Νικολακόπουλος

Giannakis

2019

IEEE Trans. Signal Process.

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Diffusion-based classifiers such as those relying on the Personalized PageRank and the Heat kernel, enjoy remarkable classification accuracy at modest computational requirements. Their performance however is affected by the extent to which the chosen diffusion captures a typically unknown label propagation mechanism, that can be specific to the underlying graph, and potentially different for each class. The present work introduces a disciplined, data-efficient approach to learning classspecific diffusion functions adapted to the underlying network topology. The novel learning approach leverages the notion of "landing probabilities" of class-specific random walks, which can be computed efficiently, thereby ensuring scalability to large graphs. This is supported by rigorous analysis of the properties of the model as well as the proposed algorithms. Furthermore, a robust version of the classifier facilitates learning even in noisy environments. Classification tests on real networks demonstrate that adapting the diffusion function to the given graph and observed labels, significantly improves the performance over fixed diffusions; reaching -and many times surpassing -the classification accuracy of computationally heavier state-of-theart competing methods, that rely on node embeddings and deep neural networks.

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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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A semi-supervised heat kernel pagerank MBO algorithm for data classification

Cited by 24 publications

References 49 publications

Graph MBO on Star Graphs and Regular Trees. With Corrections to DOI 10.1007/s00032-014-0216-8

Graph MBO on Star Graphs and Regular Trees. With Corrections to DOI 10.1007/s00032-014-0216-8

Adaptive Diffusions for Scalable Learning Over Graphs

Graphical Models in Machine Learning, Networks and Uncertainty Quantification

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