Gromov-Wasserstein Learning for Graph Matching and Node Embedding - - PowerPoint PPT Presentation

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Gromov-Wasserstein Learning for Graph Matching and Node Embedding Hongteng Xu 1 , 2 , Dixin Luo 2 , Hongyuan Zha 3 Lawrence Carin 2 1 Infinia ML, Inc. 2 Department of ECE, Duke University 3 Colledge of Computing, Georgia Tech June 13, 2019

SLIDE 1

Gromov-Wasserstein Learning for Graph Matching and Node Embedding

Hongteng Xu1,2, Dixin Luo2, Hongyuan Zha3 Lawrence Carin2

1Infinia ML, Inc. 2Department of ECE, Duke University 3Colledge of Computing, Georgia Tech

June 13, 2019

SLIDE 2

Problem Statement and Proposed Method

Given two graphs, we aim to achieve

◮ Graph matching: Finding a correspondence between their nodes. ◮ Node embedding: Embedding their nodes in the same space.

Unify them in our Gromov-Wasserstein Learning (GWL) framework. dGW (Gs, Gt) := minT∈Π(µs,µt)

i,j,i′,j′ L(cs

ij, ct i′j′)Tii′Tjj′ = minT∈Π(µs,µt)L(Cs, Ct, T), T. A B C D E 1 2 3 4 5 7 6 8 Relational matching between graphs Cost = |d(A, D) - d(1, 2)| A B C D E 1 2 3 4 5 6 7 8 Optimal transport

SLIDE 3

Gromov-Wasserstein Learning

min

Xs,Xt

min

T∈Π(µs,µt) L(Cs(Xs), Ct(Xt), T), T

Gromov-Wasserstein discrepancy

+ αK(Xs, Xt), T

Wasserstein discrepancy

+ β

k=s,t R(K(Xk, Xk), Ck)
prior information

.

A B C D E 1 2 3 4 5 6 7 8 A B C D E 1 2 3 4 5 7 6 8 Embedding space of nodes

Relational matching between graphs Cost = |d(A, D) - d(1, 2)|

Optimal transport

Update embeddings based on optimal transport and graph topology Update optimal transport based

n embeddings

and graph topology Absolute matching between graphs Cost = K(A, 1)

SLIDE 4

Experimental Results

GWD GWL-C (OT) GWL-C (Embedding) GW Discrepancy

0.0 0.1 0.2 0.3 0.4 0.5 The percentage of noisy nodes and edges 70 75 80 85 90 95 100 Node Correctness (%) 0.00050 0.00075 0.00100 0.00125 0.00150 0.00175 0.00200 GW discrepancy 0.0 0.1 0.2 0.3 0.4 0.5 The percentage of noisy nodes and edges 20 30 40 50 60 70 80 90 Node Correctness (%) 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 GW discrepancy

Table 1. Communication network matching results. Method Call→Email (Sparse) Call→Email (Dense) Node Correctness (%) Node Correctness (%) GAA 34.22 0.53 LRSA 38.20 2.93 TAME 37.39 2.67 GRAAL 39.67 0.48 MI-GRAAL 35.53 0.64 MAGNA++ 7.88 0.09 HugAlign 36.21 3.86 NETAL 36.87 1.77 GWD 23.16±0.46 1.77±0.22 GWL-R 39.64±0.57 3.80±0.23 GWL-C 40.45±0.53 4.23±0.27