A Survey on Graph Kernels

Graph kernels have become an established and widely-used technique for solving classification tasks on graphs. This survey gives a comprehensive overview of techniques for kernel-based graph classification developed in the past 15 years. We describe and categorize graph kernels based on properties inherent to their design, such as the nature of their extracted graph features, their method of computation and their applicability to problems in practice. In an extensive experimental evaluation, we study the classification accuracy of a large suite of graph kernels on established benchmarks as well as new datasets. We compare the performance of popular kernels with several baseline methods and study the effect of applying a Gaussian RBF kernel to the metric induced by a graph kernel. In doing so, we find that simple baselines become competitive after this transformation on some datasets. Moreover, we study the extent to which existing graph kernels agree in their predictions (and prediction errors) and obtain a data-driven categorization of kernels as result. Finally, based on our experimental results, we derive a practitioner's guide to kernel-based graph classification

Rule Of Thumb: Deep derotation for improved fingertip detection

We investigate a novel global orientation regression approach for articulated objects using a deep convolutional neural network. This is integrated with an in-plane image derotation scheme, DeROT, to tackle the problem of per-frame fingertip detection in depth images. The method reduces the complexity of learning in the space of articulated poses which is demonstrated by using two distinct state-of-the-art learning based hand pose estimation methods applied to fingertip detection. Significant classification improvements are shown over the baseline implementation. Our framework involves no tracking, kinematic constraints or explicit prior model of the articulated object in hand. To support our approach we also describe a new pipeline for high accuracy magnetic annotation and labeling of objects imaged by a depth camera.Comment: To be published in proceedings of BMVC 201

On the optimality of shape and data representation in the spectral domain

A proof of the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO) in representing smooth functions on surfaces is provided and adapted to the field of applied shape and data analysis. It is based on the Courant-Fischer min-max principle adapted to our case. % The theorem we present supports the new trend in geometry processing of treating geometric structures by using their projection onto the leading eigenfunctions of the decomposition of the LBO. Utilisation of this result can be used for constructing numerically efficient algorithms to process shapes in their spectrum. We review a couple of applications as possible practical usage cases of the proposed optimality criteria. % We refer to a scale invariant metric, which is also invariant to bending of the manifold. This novel pseudo-metric allows constructing an LBO by which a scale invariant eigenspace on the surface is defined. We demonstrate the efficiency of an intermediate metric, defined as an interpolation between the scale invariant and the regular one, in representing geometric structures while capturing both coarse and fine details. Next, we review a numerical acceleration technique for classical scaling, a member of a family of flattening methods known as multidimensional scaling (MDS). There, the optimality is exploited to efficiently approximate all geodesic distances between pairs of points on a given surface, and thereby match and compare between almost isometric surfaces. Finally, we revisit the classical principal component analysis (PCA) definition by coupling its variational form with a Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can handle cases that go beyond the scope defined by the observation set that is handled by regular PCA

Sequence queries on temporal graphs

Graphs that evolve over time are called temporal graphs. They can be used to describe and represent real-world networks, including transportation networks, social networks, and communication networks, with higher fidelity and accuracy. However, research is still limited on how to manage large scale temporal graphs and execute queries over these graphs efficiently and effectively. This thesis investigates the problems of temporal graph data management related to node and edge sequence queries. In temporal graphs, nodes and edges can evolve over time. Therefore, sequence queries on nodes and edges can be key components in managing temporal graphs. In this thesis, the node sequence query decomposes into two parts: graph node similarity and subsequence matching. For node similarity, this thesis proposes a modified tree edit distance that is metric and polynomially computable and has a natural, intuitive interpretation. Note that the proposed node similarity works even for inter-graph nodes and therefore can be used for graph de-anonymization, network transfer learning, and cross-network mining, among other tasks. The subsequence matching query proposed in this thesis is a framework that can be adopted to index generic sequence and time-series data, including trajectory data and even DNA sequences for subsequence retrieval. For edge sequence queries, this thesis proposes an efficient storage and optimized indexing technique that allows for efficient retrieval of temporal subgraphs that satisfy certain temporal predicates. For this problem, this thesis develops a lightweight data management engine prototype that can support time-sensitive temporal graph analytics efficiently even on a single PC

쌍별 색 개선과 효율적인 백트래킹을 이용한 빠른 그래프 동형 알고리즘

학위논문(박사) -- 서울대학교대학원 : 공과대학 컴퓨터공학부, 2021.8. 구건모.Graph isomorphism is a core problem in graph analysis of various domains including social networks, bioinformatics, chemistry, and so on. As real-world graphs are getting bigger and bigger, applications demand practically fast algorithms that can run on large-scale graphs. Existing approaches, however, show limited performances on large-scale real-world graphs either in time or space. Also, graph isomorphism query processing is often required in many applications, which is a natural generalization of graph isomorphism for multiple graphs. In this thesis we present fast algorithms for graph isomorphism and graph isomorphism query processing. First, we present a new approach to graph isomorphism, which is the framework of pairwise color refinement and efficient backtracking. Within the framework, we introduce three efficient techniques, which together lead to a much faster and scalable algorithm for graph isomorphism. Experiments on real-world datasets show that our algorithm outperforms state-of-the-art solutions by up to several orders of magnitude in terms of running time. Second, We develop an efficient algorithm for graph isomorphism query processing. We use a two-level index using degree sequences and color-label distributions. Experimental results on real datasets show that our algorithm is orders of magnitude faster than the state-of-the-art algorithms in terms of index construction time, and it runs faster than existing algorithms in terms of query processing time as the graph sizes increase.그래프 동형 문제는 소셜 네트워크 서비스, 생물정보학, 화학정보학 등등 다양한 응용 분야에서 그래프 분석을 위해 다루고 있는 핵심 문제이다. 실생활에서 다루는 그래프 데이터의 크기가 커져 감에 따라, 대용량의 그래프를 처리할 수 있는 그래프 동형 알고리즘의 필요성이 높아지고 있다. 그러나 현재 존재하는 그래프 동형 알고리즘들은 대용량의 그래프에 대해서 시간 혹은 공간 측면에서 한계를 보여준다. 응용 분야 중에서는 여러 개의 그래프들 중에서 하나의 쿼리 그래프와 동형인 그래프를 모두 찾는 문제, 즉 그래프 동형 쿼리 프로세싱을 종종 요구하기도 한다. 본 논문에서는 대용량의 실제 그래프 데이터에 대해서 그래프 동형 문제와 그래프 동형 쿼리 프로세싱 문제를 빠르게 푸는 알고리즘들을 제안한다. 첫 번째로, 본 논문에서는 그래프 동형 문제를 위한 빠르고 확장성 있는 알고리즘을 제안한다. 이를 위해 쌍별 색 개선(pairwise color refinement)과 효율적인 백트래킹으로 구성된 프레임워크를 소개한다. 이 프레임워크 내에서 세 가지 효율적인 테크닉을 사용한다. 실제 그래프 데이터에 대한 실험을 통해 본 알고리즘이 현존하는 가장 빠른 알고리즘들보다 평균 수천 배 빠름을 보였다. 두 번째로, 본 논문에서는 그래프 동형 쿼리 프로세싱을 위한 효율적인 알고리즘을 개발한다. 본 알고리즘은 차수열과 색-레이블 분포를 이용한 인덱스를 이용한다. 실제 그래프 데이터에 대한 실험을 통해 본 알고리즘이 현존하는 알고리즘들보다 인덱싱 시간에서는 항상 평균 수천 배 빠르고, 쿼리 처리 시간에서는 중$\cdot$대용량의 그래프들에 대해서 평균 수십 배 빠르게 동작하는 것을 보였다.1. Introduction 1 1.1. Background 1 1.2. Organization 3 2. Preliminaries 4 2.1. Notation 4 2.2. Problem Definitions 6 2.3. Related Work 7 3. Graph Isomorphism 9 3.1. Algorithm Overview 12 3.2. Pairwise Color Refinement and Binary Cell Mapping 13 3.3. Compressed Candidate Space 16 3.4. Backtracking and Partial Failing Sets 21 3.5. Performance Evaluation 31 3.5.1. Comparing with Existing Solutions 35 3.5.2. Effectiveness of Individual Techniques 39 3.5.3. Analysis with Varying Degrees of Similarity 42 3.5.4. Sensitivity Analysis 46 4. Graph Isomorphism Query Processing 48 4.1. Canonical Coloring 51 4.2. Index Construction 56 4.3. Query Processing 59 4.4. Performance Evaluation 63 4.4.1. Varying Number of Hops 67 4.4.2. Varying Number of Data Graphs 74 5. Conclusion 78 5.1. Summary 78 5.2. Future Directions 79 요약 95박