[논문 리뷰] Wasserstein Weisfeiler-Lehman Graph Kernels

Matteo Togninalli, Elisabetta Ghisu외 3명|arXiv (Cornell University)|2019. 06. 04.

Graph Theory and Algorithms참고 문헌 43인용 수 62

한 줄 요약

WWL은 Weisfeiler–Lehman 임베딩과 Wasserstein 거리를 결합하여 노드 특성 분포로 그래프를 비교하며, 연속 속성에서 강한 성능과 범주형 라벨에서의 경쟁력을 달성한다.

ABSTRACT

Most graph kernels are an instance of the class of $\mathcal{R}$-Convolution kernels, which measure the similarity of objects by comparing their substructures. Despite their empirical success, most graph kernels use a naive aggregation of the final set of substructures, usually a sum or average, thereby potentially discarding valuable information about the distribution of individual components. Furthermore, only a limited instance of these approaches can be extended to continuously attributed graphs. We propose a novel method that relies on the Wasserstein distance between the node feature vector distributions of two graphs, which allows to find subtler differences in data sets by considering graphs as high-dimensional objects, rather than simple means. We further propose a Weisfeiler-Lehman inspired embedding scheme for graphs with continuous node attributes and weighted edges, enhance it with the computed Wasserstein distance, and thus improve the state-of-the-art prediction performance on several graph classification tasks.

연구 동기 및 목표

• 간단한 합/평균보다 서브구조 분포를 더 잘 포착하는 그래프 커널을 동기 부여한다.
• 노드 특징 집합 간의 그래프 Wasserstein 거리 (GWD)를 도입한다.
• 연속 속성 및 가중치가 있는 간선을 다루는 WL에서 영감을 받은 임베딩 방식을 개발한다.
• 개선된 그래프 분류를 위해 GWD를 Laplacian 유사 WWL 커널과 결합한다.
• 범주형 라벨과 연속 속성 모두를 가진 벤치마크에서 WWL을 평가한다.]
• method:["각 그래프에 대해 노드 임베딩을 출력하는 그래프 임베딩 스킴 f를 정의한다.","Graph Wasserstein Distance D^f_W(G,G')를 W1(f(G),f(G'))로 정의한다.","가중치를 반영하는 평균 전파 단계를 통해 WL을 연속 속성에 확장한다.","Lapla cian 유사 커널로서 K_WWL = exp(-λ D^f_W)를 구성한다.","범주형 WWL(양의 정의성)과 연속 WWL(이론적으로 부정적 정의성을 가지며 Kreĭn-SVM으로 다룬다.)를 구분한다.","큰 그래프에서 Wasserstein 계산 속도를 높이기 위해 선택적으로 Sinkhorn 정규화를 사용한다."]
• research_questions:[
• Can Wasserstein distance between node-embedding distributions capture finer graph similarities than traditional substructure aggregations?
• How can WL-like embeddings be extended to continuously attributed graphs with weighted edges?
• Does WWL improve graph classification on benchmarks with continuous node attributes while remaining competitive on categorical-label datasets?

제안 방법

• Define graph embedding scheme f that outputs node embeddings for each graph.
• Define Graph Wasserstein Distance D^f_W(G,G') as W1(f(G),f(G')).
• Extend WL to continuous attributes via an averaging propagation step accounting for edge weights.
• Construct WWL kernel K_WWL = exp(-λ D^f_W) as a Laplacian-like kernel.
• Differentiate categorical WWL (positive definite) from continuous WWL (이론적으로 부정적 정의성을 가지며 Kreĭn-SVM으로 다룬다).
• Optionally use Sinkhorn regularisation to speed up Wasserstein computations for large graphs.

실험 결과

연구 질문

• RQ1 Can Wasserstein distance between node-embedding distributions capture finer graph similarities than traditional substructure aggregations?
• RQ2 How can WL-like embeddings be extended to continuously attributed graphs with weighted edges?
• RQ3 Does WWL improve graph classification on benchmarks with continuous node attributes while remaining competitive on categorical-label datasets?

주요 결과

• WWL matches state-of-the-art on categorically labelled graphs (comparable to WL-OA).
• WWL significantly outperforms baselines on graphs with continuous node attributes across several datasets.
• The categorical WWL kernel is proven positive definite for all λ>0.
• WWL achieves top average ranking in continuous-attribute experiments across multiple datasets.
• RBF-WL baselines and other graph kernels are outperformed by WWL on several continuous-attribute tasks.
• WWL benefits from using optimal transport to compare whole distributions of node features rather than mere aggregates.