Multi-attributed Graph Matching with Multi-layer Random Walks

Graph matching aims at establishing correspondences between graph elements, and is widely used in many computer vision tasks. Among recently proposed graph matching algorithms, those utilizing the path following strategy have attracted special research attentions due to their exhibition of state-of-the-art performances. However, the paths computed in these algorithms often contain singular points, which could hurt the matching performance if not dealt properly. To deal with this issue, we propose a novel path following strategy, named branching path following (BPF), to improve graph matching accuracy. In particular, we first propose a singular point detector by solving a KKT system, and then design a branch switching method to seek for better paths at singular points. Moreover, to reduce the computational burden of the BPF strategy, an adaptive path estimation (APE) strategy is integrated into BPF to accelerate the convergence of searching along each path. A new graph matching algorithm named ABPF-G is developed by applying APE and BPF to a recently proposed path following algorithm named GNCCP (Liu & Qiao 2014). Experimental results reveal how our approach consistently outperforms state-of-the-art algorithms for graph matching on five public benchmark datasets.

Graph and hypergraph matching are fundamental problems in computer vision, which have been receiving a steadily growing attention in the last two decades due to the recent modern algorithms and models that are both fast and accurate. Graph and hypergraph matching are successful in tackling many real-world tasks that require 2D and 3D feature matching and alignment. Matching is a general problem in vision and graphs have always been quintessential for modeling information that has both local and longer range connections. Therefore, graph matching will continue to influence computer vision and robotics research, crossing over many different mathematical and computational models, including Bayesian networks, conditional and Markov random fields, spectral graph models, and, more recently, deep neural networks. While graph matching is limited to using pairwise relationships, hypergraph matching permits the use of relationships between sets of features of any order. Consequently, it carries the promise to make matching more robust to changes in scale, deformations, and outliers. In this chapter, we present several key ideas, computational models, and optimization methods that introduced in the literature the possibility to learn graph and hypergraph matching in a semi-supervised or even completely unsupervised fashion. We start by describing two methods for solving graph and hypergraph matching based on spectral matching and integer projected fixed point methods, introduced in Chap. 1. Then, based on the same intuition (also discussed in Chap. 1) that correct assignments form a strong cluster of agreements, we follow up by introducing the main approach to learning as it applies to graph and hypergraph matching, as well as the more general Maximum A Posteriori (MAP) inference problem in probabilistic graphical models (MRFs and CRFs). In the last part of the chapter, we demonstrate experimentally the effectiveness of the algorithms presented and, whenever possible, provide extensive comparisons to other top published methods. Moreover, we show that our learning approach is general and can significantly boost the performance of different state-of-the-art matching algorithms in the literature.

Many graph matching algorithms follow the approach of node-similarity measurement, that is, matching graphs by means of comparing the corresponding pairs of nodes of the graphs. Based on this idea, the authors propose a high-level schema for node-to-node graph matching, namely N2N graph matching algorithm schema. The chapter shows that such a N2N graph matching algorithm schema is versatile enough to subsume most of the representative node-to-node based graph matching algorithms. It is also shown that improved algorithms can be derived from this N2N graph matching schema, compared with various corresponding algorithms. In addition, the authors point out the limitation and constraints of the propose algorithm schema and suggest some possible treatments.

Software tools for Social Network Analysis (SNA) are being developed which support various types of analysis of social networks extracted from social media websites (e.g., Twitter). Once extracted and stored in a database such social networks are amenable to analysis by SNA software. This data analysis often involves searching for occurrences of various subgraph patterns (i.e., graphical representations of entities and relationships). The authors have developed the Graph Matching Toolkit (GMT) which provides an intuitive Graphical User Interface (GUI) for a heuristic graph matching algorithm called the Truncated Search Tree (TruST) algorithm. GMT is a visual interface for graph matching algorithms processing large social networks. GMT enables an analyst to draw a subgraph pattern by using a mouse to select categories and labels for nodes and links from drop-down menus. GMT then executes the TruST algorithm to find the top five occurrences of the subgraph pattern within the social network stored in the database. GMT was tested using a simulated counter-insurgency dataset consisting of cellular phone communications within a populated area of operations in Iraq. The results indicated GMT (when executing the TruST graph matching algorithm) is a time-efficient approach to searching large social networks. GMT’s visual interface to a graph matching algorithm enables intelligence analysts to quickly analyze and summarize the large amounts of data necessary to produce actionable intelligence.

Graph matching based on spectral embedding with missing value

Quadratic assignment problems arise in a wide variety of domains, spanning operations research, graph theory, computer vision, and neuroscience, to name a few. The graph matching problem is a special case of the quadratic assignment problem, and graph matching is increasingly important as graph-valued data is becoming more prominent. With the aim of efficiently and accurately matching the large graphs common in big data, we present our graph matching algorithm, the Fast Approximate Quadratic assignment algorithm. We empirically demonstrate that our algorithm is faster and achieves a lower objective value on over 80% of the QAPLIB benchmark library, compared with the previous state-of-the-art. Applying our algorithm to our motivating example, matching C. elegans connectomes (brain-graphs), we find that it efficiently achieves performance.

A graph matching algorithm based on concavely regularized convex relaxation

We propose a method to perform active graph matching in which the active learner queries one of the nodes of the first graph and the oracle feedback is the corresponding node of the other graph. The method uses any graph matching algorithm that iteratively updates a probability matrix between nodes (Graduated Assignment, Expectation Maximisation or Probabilistic Relaxation). The oracle's feedback is used to update the costs between nodes and arcs of both graphs. We present and validate four different active strategies based on the probability matrix between nodes. It is not needed to modify the code of the graph-matching algorithms, since our method simply needs to read the probability matrix and to update the costs between nodes and arcs. Practical validation shows that with few oracle's feedbacks, the algorithm finds the labelling that the user considers optimal because imposing few labellings the other ones are corrected automatically.

This paper introduces a novel algorithm for performing Attributed Graph Matching (AGM). A salient characteristic of the Interpolator- Based Kronecker Product Graph Matching (IBKPGM) algorithm is that it does not require the explicit calculation of compatibility values between vertices and edges, either using compatibility functions or probability distributions. No assumption is made about the adjacency structure of the graphs to be matched. The IBKPGM algorithm uses Reproducing Kernel Hilbert Space (RKHS) interpolator theory to obtain an unconstrained estimate to the Kronecker Match Matrix (KMM) from which a permutation sub-matrix is inferred.

Kronecker product graph matching

The graph is an essential data structure for representing relational information. When graphs are used to represent objects, comparing objects amounts to graph matching. Inexact graph matching is the process of finding the best possible matching between two graphs when exact matching is impossible. We propose a new algorithm for the inexact matching problem. The new algorithm decomposes the matching process into K phases, where the value of K ranges from 1 to the minimum of the numbers of nodes in the two graphs to be matched. The efficiency of the new algorithm results from the use of small values of K, significantly reducing the search space while still producing very good matchings (most of them optimal) between graphs. The algorithm is compared with the error-correcting subgraph isomorphism algorithm based on A*.

Most of the approaches which address the problem of 3D object retrieval, use global descriptors of the objects which fail to consistently compensate for the intra-class variability of articulated objects. In this paper, a retrieval methodology is presented which is based upon a graph-based object representation. This is composed of a meaningful new mesh segmentation along with a graph matching between the graph of the query object and each of the graphs that correspond to the objects of the 3D object database. The graph matching algorithm is based on the Earth Mover's Distance (EMD) similarity measure which is calculated using a new ground distance assignment. The superior performance of the proposed methodology is shown after an extensive experimentation comprising alternative descriptors for the constituent components of the 3D object as well as comparison with state of the art retrieval algorithms.

We describe a novel RGBD relocalisation algorithm based on key point matching. It combines two components. First, a graph matching algorithm which takes into account the pairwise 3-D geometry amongst the key points, giving robust relocalisation. Second, a point selection process which provides an even distribution of the ‘most matchable’ points across the scene based on non-maximum suppression within voxels of a volumetric grid. This ensures a bounded set of matchable key points which enables tractable and scalable graph matching at frame rate. We present evaluations using a public dataset and our own more difficult dataset containing large pose changes, fast motion and non-stationary objects. It is shown that the method significantly out performs state-of-the-art methods.

Measures of vertex similarity have been incorporated in graph matching algorithms. Graph matching tries to retrieve a 1-1 correspondence between vertices of two given graphs. In this paper, the vertex similarity measure of Blondel et al. is studied for its usefulness in detecting graph isomorphism. Firstly, the applicability of this measure to distinguish similar pairs from dissimilar pairs is shown to be limited in scope even for small graphs. In a preliminary experiment, we show that Blondel’s vertex similarity measure does not retrieve the isomorphism within a graph of 14 nodes. We propose a refinement of Blondel’s measure. Zager et al. also refine Blondel’s measure and further propose a graph matching algorithm. We propose a graph matching algorithm based on the lines of Zager et al. and test our algorithm against Zager’s as well as Blondel’s and show that the proposed refinement performs better than both the measures with regard to graph isomorphism problem. The performance is evaluated systematically on a large bench mark data set made available by Foggia et al. The proposed algorithm performs with 90.10% accuracy on all of the 18,200 pairs of isomorphic graphs available in the benchmark dataset.Keywordsvertex similaritygraph isomorphismgraph matchinglarge graphs

How can we find the virtual twin (i.e., the same or similar user) on Linked In for a user on Facebook? How can we effectively link an information network with a social network to support cross-network search? Graph alignment - the task of finding the node correspondences between two given graphs - is a fundamental building block in numerous application domains, such as social networks analysis, bioinformatics, chemistry, pattern recognition. In this work, we focus on aligning bipartite graphs, a problem which has been largely ignored by the extensive existing work on graph matching, despite the ubiquity of those graphs (e.g., users-groups network). We introduce a new optimization formulation and propose an effective and fast algorithm to solve it. We also propose a fast generalization of our approach to align unipartite graphs. The extensive experimental evaluations show that our method outperforms the state-of-art graph matching algorithms in both alignment accuracy and running time, being up to 10x more accurate or 174x faster on real graphs.