We propose an unsupervised object matching method for relational data, which finds matchings between objects in different relational datasets without correspondence information. For example, the proposed method matches documents in different languages in multi-lingual document-word networks without dictionaries nor alignment information. The proposed method assumes that each object has latent vectors, and the probability of neighbor objects is modeled by the inner-product of the latent vectors, where the neighbors are generated by short random walks over the relations. The latent vectors are estimated by maximizing the likelihood of the neighbors for each dataset. The estimated latent vectors contain hidden structural information of each object in the given relational dataset. Then, the proposed method linearly projects the latent vectors for all the datasets onto a common latent space shared across all datasets by matching the distributions while preserving the structural information. The projection matrix is estimated by minimizing the distance between the latent vector distributions with an orthogonality regularizer. To represent the distributions effectively, we use the kernel embedding of distributions that hold high-order moment information about a distribution as an element in a reproducing kernel Hilbert space, which enables us to calculate the distance between the distributions without density estimation. The structural information encoded in the latent vectors are preserved by using the orthogonality regularizer. We demonstrate the effectiveness of the proposed method with experiments using real-world multi-lingual document-word relational datasets and multiple user-item relational datasets.
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@cite_53: As a fundamental problem in computer vision, graph matching problem can usually be formulated as a Quadratic Programming (QP) problem with doubly stochastic and discrete (integer) constraints. Since it is NP-hard, approximate algorithms are required. In this paper, we present a new algorithm, called Multiplicative Update Graph Matching (MPGM), that develops a multiplicative update technique to solve the QP matching problem. MPGM has three main benefits: (1) theoretically, MPGM solves the general QP problem with doubly stochastic constraint naturally whose convergence and KKT optimality are guaranteed. (2) Em- pirically, MPGM generally returns a sparse solution and thus can also incorporate the discrete constraint approximately. (3) It is efficient and simple to implement. Experimental results show the benefits of MPGM algorithm.
@cite_54: In this paper, we propose a general framework for graph matching which is suitable for different problems of pattern recognition. The pattern representation we assume is at the same time highly structured, like for classic syntactic and structural approaches, and of subsymbolic nature with real-valued features, like for connectionist and statistic approaches. We show that random walk based models, inspired by Google's PageRank, give rise to a spectral theory that nicely enhances the graph topological features at node level. As a straightforward consequence, we derive a polynomial algorithm for the classic graph isomorphism problem, under the restriction of dealing with Markovian spectrally distinguishable graphs (MSD), a class of graphs that does not seem to be easily reducible to others proposed in the literature. The experimental results that we found on different test-beds of the TC-15 graph database show that the defined MSD class "almost always" covers the database, and that the proposed algorithm is significantly more efficient than top scoring VF algorithm on the same data. Most interestingly, the proposed approach is very well-suited for dealing with partial and approximate graph matching problems, derived for instance from image retrieval tasks. We consider the objects of the COIL-100 visual collection and provide a graph-based representation, whose node's labels contain appropriate visual features. We show that the adoption of classic bipartite graph matching algorithms offers a straightforward generalization of the algorithm given for graph isomorphism and, finally, we report very promising experimental results on the COIL-100 visual collection.
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@cite_52: In this paper we address the problem of matching two 3D shapes by representing them using the eigenvalues and eigenvectors of the discrete diffusion operator. This provides representation framework useful for scale-space local shape descriptors and shape comparisons. We formally introduce diffusion embedding and we propose unit hypersphere normalizations of this embedding. We also propose a method to find compatible time scale while matching two different shapes with varying size and sampling. We propose a practical algorithm that seeks the largest set of mutually consistent point-to-point matches between two shapes based on isometric consistency between the two embedded shapes. We illustrate our method with several examples of matching shapes at various scales.
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@cite_59: Graph matching is an essential problem in computer vision and machine learning. In this paper, we introduce a random walk view on the problem and propose a robust graph matching algorithm against outliers and deformation. Matching between two graphs is formulated as node selection on an association graph whose nodes represent candidate correspondences between the two graphs. The solution is obtained by simulating random walks with reweighting jumps enforcing the matching constraints on the association graph. Our algorithm achieves noise-robust graph matching by iteratively updating and exploiting the confidences of candidate correspondences. In a practical sense, our work is of particular importance since the real-world matching problem is made difficult by the presence of noise and outliers. Extensive and comparative experiments demonstrate that it outperforms the state-of-the-art graph matching algorithms especially in the presence of outliers and deformation.
@cite_2: We present DeepWalk, a novel approach for learning latent representations of vertices in a network. These latent representations encode social relations in a continuous vector space, which is easily exploited by statistical models. DeepWalk generalizes recent advancements in language modeling and unsupervised feature learning (or deep learning) from sequences of words to graphs. DeepWalk uses local information obtained from truncated random walks to learn latent representations by treating walks as the equivalent of sentences. We demonstrate DeepWalk's latent representations on several multi-label network classification tasks for social networks such as BlogCatalog, Flickr, and YouTube. Our results show that DeepWalk outperforms challenging baselines which are allowed a global view of the network, especially in the presence of missing information. DeepWalk's representations can provide F1 scores up to 10 higher than competing methods when labeled data is sparse. In some experiments, DeepWalk's representations are able to outperform all baseline methods while using 60 less training data. DeepWalk is also scalable. It is an online learning algorithm which builds useful incremental results, and is trivially parallelizable. These qualities make it suitable for a broad class of real world applications such as network classification, and anomaly detection.
@cite_47: In this paper we propose an inexact spectral matching algorithm that embeds large graphs on a low-dimensional isometric space spanned by a set of eigenvectors of the graph Laplacian. Given two sets of eigenvectors that correspond to the smallest non-null eigenvalues of the Laplacian matrices of two graphs, we project each graph onto its eigenenvectors. We estimate the histograms of these one-dimensional graph projections (eigenvector histograms) and we show that these histograms are well suited for selecting a subset of significant eigenvectors, for ordering them, for solving the sign-ambiguity of eigenvector computation, and for aligning two embeddings. This results in an inexact graph matching solution that can be improved using a rigid point registration algorithm. We apply the proposed methodology to match surfaces represented by meshes.
@cite_16: We present a method for learning max-weight matching predictors in bipartite graphs. The method consists of performing maximum a posteriori estimation in exponential families with sufficient statistics that encode permutations and data features. Although inference is in general hard, we show that for one very relevant application - web page ranking - exact inference is efficient. For general model instances, an appropriate sampler is readily available. Contrary to existing max-margin matching models, our approach is statistically consistent and, in addition, experiments with increasing sample sizes indicate superior improvement over such models. We apply the method to graph matching in computer vision as well as to a standard benchmark dataset for learning web page ranking, in which we obtain state-of-the-art results, in particular improving on max-margin variants. The drawback of this method with respect to max-margin alternatives is its runtime for large graphs, which is comparatively high.
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@cite_25: Prediction tasks over nodes and edges in networks require careful effort in engineering features used by learning algorithms. Recent research in the broader field of representation learning has led to significant progress in automating prediction by learning the features themselves. However, present feature learning approaches are not expressive enough to capture the diversity of connectivity patterns observed in networks. Here we propose node2vec, an algorithmic framework for learning continuous feature representations for nodes in networks. In node2vec, we learn a mapping of nodes to a low-dimensional space of features that maximizes the likelihood of preserving network neighborhoods of nodes. We define a flexible notion of a node's network neighborhood and design a biased random walk procedure, which efficiently explores diverse neighborhoods. Our algorithm generalizes prior work which is based on rigid notions of network neighborhoods, and we argue that the added flexibility in exploring neighborhoods is the key to learning richer representations. We demonstrate the efficacy of node2vec over existing state-of-the-art techniques on multi-label classification and link prediction in several real-world networks from diverse domains. Taken together, our work represents a new way for efficiently learning state-of-the-art task-independent representations in complex networks.
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The proposed method is related to graph matching and network alignment methods, which find correspondence so that matched objects have the same edge relation and or the similar attributes @cite_13 @cite_16 @cite_60 @cite_28 @cite_53 @cite_57 @cite_33 @cite_59 @cite_54 @cite_52 . The proposed method is different from them, in a way that the proposed method considers not only directly connected edge relation but also the intrinsic hidden structure of the given relational dataset without attributes by embedding objects onto a latent space. The proposed method is based on the recent advancements of graph representation learning methods, such as DeepWalk @cite_2 , LINE @cite_29 and node2vec @cite_25 . These methods obtain node representations that contain structural information in the graph, and have been successfully used in link prediction and node classification. Although spectral methods for graph matching have been proposed @cite_47 @cite_17 , DeepWalk and node2vec achieved better performance than the spectral methods in graph representation learning @cite_2 @cite_25 .