Non-bayesian Graph Matching without Explicit Compatibility Calculations

This paper introduces a novel algorithm for performing Attributed Graph Matching (AGM). The Orthonormal Kernel Kronecker Product Graph Matching (OKKPGM) algorithm is based on the recently introduced Kronecker Product Graph Matching (KPGM) formulation. However, unlike previous algorithms based on the KPGM formulation which avoided explicit compatibility calculations, the OKKPGM algorithm obtains an estimate to the Kronecker Match Matrix (KMM) using kernel function evaluations. A permutation sub-matrix (match matrix), is then inferred from the KMM using a summation procedure.

This paper addresses the multi-attributed graph matching problem considering multiple attributes jointly while preserving the characteristics of each attribute. Since most of conventional graph matching algorithms integrate multiple attributes to construct a single attribute in an oversimplified way, the information from multiple attributes are not often fully exploited. In order to solve this problem, we propose a novel multi-layer graph structure that can preserve the particularities of each attribute in separated layers. Then, we also propose a multi-attributed graph matching algorithm based on the random walk centrality for the proposed multi-layer graph structure. We compare the proposed algorithm with other state-of-the-art graph matching algorithms based on the single-layer structure using synthetic and real datasets, and prove the superior performance of the proposed multi-layer graph structure and matching algorithm.

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.

Kronecker product graph matching

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 matching based on spectral embedding with missing value

This paper presents a multi-layer reproducing kernel Hilbert space (RKHS) approach for probability distribution to real and probability distribution to function regressions. The approach maps the distributions into RKHS by distribution embeddings and, then, constructs a multi-layer RKHS within which the multi-kernel distribution regression can be implemented using an existing kernel regression algorithm, such as kernel recursive least squares (KRLS). The numerical simulations on synthetic data obtained via Gaussian mixtures show that the proposed approach outperforms existing probability distribution (DR) regression algorithms by achieving smaller mean squared errors (MSEs) and requiring less training samples.

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.

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.

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.

Similarity measurement of two probability distributions is important in many applications of statistics. Embedding such distributions into a reproducing kernel Hilbert space (RKHS) has many favorable properties. The choice of the reproducing kernel is crucial in the approach. We study this question by considering the similarity of two distributions of the same class. In particular, we investigate when the RKHS embedding is “admissible” in the sense that the distance between the embeddings should become smaller when the expectations are getting closer or when the variance is increasing to infinity. We give conditions on the widely-used translation-invariant reproducing kernels to be admissible. We also extend the study to multivariate non-symmetric Gaussian distributions.

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.

A graph matching algorithm based on concavely regularized convex relaxation

As a fundamental problem in pattern recognition, graph matching has applications in a variety of fields, from computer vision to computational biology. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. Many formulations of this problem can be cast in general as a quadratic assignment problem, where a linear term in the objective function encodes node compatibility and a quadratic term encodes edge compatibility. The main research focus in this theme is about designing efficient algorithms for approximately solving the quadratic assignment problem, since it is NP-hard. In this paper we turn our attention to a different question: how to estimate compatibility functions such that the solution of the resulting graph matching problem best matches the expected solution that a human would manually provide. We present a method for learning graph matching: the training examples are pairs of graphs and the 'labels' are matches between them. Our experimental results reveal that learning can substantially improve the performance of standard graph matching algorithms. In particular, we find that simple linear assignment with such a learning scheme outperforms Graduated Assignment with bistochastic normalisation, a state-of-the-art quadratic assignment relaxation algorithm.

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*.