Wasserstein-based Graph Alignment

Maretić, Hermina Petric; Gheche, Mireille El; Minder, Matthias; Chierchia, Giovanni; Frossard, Pascal

doi:10.48550/arxiv.2003.06048

Cited by 4 publications

(8 citation statements)

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“…Spectral methods. One line of work [23,24,12] uses graph spectral techniques to define OT problems for graphs. In particular, this approach associates to each graph a multivariate Gaussian with zero mean and covariance matrix equal to the pseudoinverse of the graph Laplacian.…”

Section: Related Workmentioning

confidence: 99%

“…The Wasserstein distance between Gaussians in the same space may be computed analytically in terms of the respective covariance matrices. For graphs with different numbers of vertices, [24] and [12] propose to optimize this distance over soft many-to-one assignments between vertices in either graph. At present, this family of approaches is unable to incorporate available feature information or underlying cost functions, relying completely on intrinsic structure in their respective optimization problems.…”

Section: Related Workmentioning

confidence: 99%

“…Intuitively, the topology of a graph with block structure will depend more strongly on edges between blocks than within blocks and thus we regard G 2 as more dissimilar to G 1 than G 3 is to G 1 . In Table 1, we provide the distances computed by GraphOTC for two different costs, as well as distances computed by two other graph OT methods known as GW [24] and GOT [23], respectively. We find that GraphOTC and GOT regard G 2 as more dissimilar from G 1 than G 3 is.…”

Section: Stochastic Block Modelsmentioning

confidence: 99%

“…Examples of applications include image captioning [8], object recognition [38], domain adaptation [22], aligning single-cell multi-omics data [11], and language comparison [4]. Some recent work [29,32,23,24,12] addresses the problems of graph comparison and alignment using techniques from optimal transport. In the optimal transport (OT) problem, one seeks a plan for transporting mass between two probability measures of interest that minimizes expected cost.…”

Section: Introductionmentioning

confidence: 99%

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Graph Optimal Transport with Transition Couplings of Random Walks

Yi¹,

O’Connor²,

McGoff³

et al. 2021

Preprint

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We present a novel approach to optimal transport between graphs from the perspective of stationary Markov chains. A weighted graph may be associated with a stationary Markov chain by means of a random walk on the vertex set with transition distributions depending on the edge weights of the graph. After drawing this connection, we describe how optimal transport techniques for stationary Markov chains may be used in order to perform comparison and alignment of the graphs under study. In particular, we propose the graph optimal transition coupling problem, referred to as GraphOTC, in which the Markov chains associated to two given graphs are optimally synchronized to minimize an expected cost. The joint synchronized chain yields an alignment of the vertices and edges in the two graphs, and the expected cost of the synchronized chain acts as a measure of distance or dissimilarity between the two graphs. We demonstrate that GraphOTC performs equal to or better than existing state-of-the-art techniques in graph optimal transport for several tasks and datasets. Finally, we also describe a generalization of the GraphOTC problem, called the FusedOTC problem, from which we recover the GraphOTC and OT costs as special cases.Preprint. Under review.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Stochastic Block Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Graph Optimal Transport with Transition Couplings of Random Walks

Yi¹,

O’Connor²,

McGoff³

et al. 2021

Preprint

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“…To capture global graph structure, Maretic et al [23] proposed a Wasserstein distance between graph signal distributions by resorting to graph Laplacian matrices. This method was initially constrained to graphs of the same sizes, but recently extended to graphs of different sizes by formulating graph matching as a one-to-many assignment problem [24]. Xu et al [51] proposed to jointly align graphs and learn node embeddings using a Gromov-Wasserstein distance.…”

Section: Related Workmentioning

confidence: 99%

A Regularized Wasserstein Framework for Graph Kernels

Wijesinghe¹,

Wang²,

Gould³

2021

Preprint

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We propose a learning framework for graph kernels, which is theoretically grounded on regularizing optimal transport. This framework provides a novel optimal transport distance metric, namely Regularized Wasserstein (RW) discrepancy, which can preserve both features and structure of graphs via Wasserstein distances on features and their local variations, local barycenters and global connectivity. Two strongly convex regularization terms are introduced to improve the learning ability. One is to relax an optimal alignment between graphs to be a cluster-tocluster mapping between their locally connected vertices, thereby preserving the local clustering structure of graphs. The other is to take into account node degree distributions in order to better preserve the global structure of graphs. We also design an efficient algorithm to enable a fast approximation for solving the optimization problem. Theoretically, our framework is robust and can guarantee the convergence and numerical stability in optimization. We have empirically validated our method using 12 datasets against 16 state-of-the-art baselines. The experimental results show that our method consistently outperforms all stateof-the-art methods on all benchmark databases for both graphs with discrete attributes and graphs with continuous attributes.

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fGOT: Graph Distances Based on Filters and Optimal Transport

Maretić

Gheche²,

Chierchia

et al. 2022

AAAI

Self Cite

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Graph comparison deals with identifying similarities and dissimilarities between graphs. A major obstacle is the unknown alignment of graphs, as well as the lack of accurate and inexpensive comparison metrics. In this work we introduce the filter graph distance. It is an optimal transport based distance which drives graph comparison through the probability distribution of filtered graph signals. This creates a highly flexible distance, capable of prioritising different spectral information in observed graphs, offering a wide range of choices for a comparison metric. We tackle the problem of graph alignment by computing graph permutations that minimise our new filter distances, which implicitly solves the graph comparison problem. We then propose a new approximate cost function that circumvents many computational difficulties inherent to graph comparison and permits the exploitation of fast algorithms such as mirror gradient descent, without grossly sacrificing the performance. We finally propose a novel algorithm derived from a stochastic version of mirror gradient descent, which accommodates the non-convexity of the alignment problem, offering a good trade-off between performance accuracy and speed. The experiments on graph alignment and classification show that the flexibility gained through filter graph distances can have a significant impact on performance, while the difference in speed offered by the approximation cost makes the framework applicable in practical settings.

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Wasserstein-based Graph Alignment

Cited by 4 publications

References 0 publications

Graph Optimal Transport with Transition Couplings of Random Walks

Graph Optimal Transport with Transition Couplings of Random Walks

A Regularized Wasserstein Framework for Graph Kernels

fGOT: Graph Distances Based on Filters and Optimal Transport

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