This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the Erd\H{o}s-R\'enyi model where the two graphs are subsampled from a common parent Erd\H{o}s-R\'enyi graph $\mathcal{G}(n,p)$. For dense graphs with $p=n^{-o(1)}$, we prove that there exists a sharp threshold, above which one can correctly match all but a vanishing fraction of vertices and below which correctly matching any positive fraction is impossible, a phenomenon known as the "all-or-nothing" phase transition. Even more strikingly, in the Gaussian setting, above the threshold all vertices can be exactly matched with high probability. In contrast, for sparse Erd\H{o}s-R\'enyi graphs with $p=n^{-\Theta(1)}$, we show that the all-or-nothing phenomenon no longer holds and we determine the thresholds up to a constant factor. Along the way, we also derive the sharp threshold for exact recovery, sharpening the existing results in Erd\H{o}s-R\'enyi graphs. The proof of the negative results builds upon a tight characterization of the mutual information based on the truncated second-moment computation and an "area theorem" that relates the mutual information to the integral of the reconstruction error. The positive results follows from a tight analysis of the maximum likelihood estimator that takes into account the cycle structure of the induced permutation on the edges.
We contribute to approximate algorithms for the quadratic assignment problem also known as graph matching. Inspired by the success of the fusion moves technique developed for multilabel discrete Markov random fields, we investigate its applicability to graph matching. In particular, we show how fusion moves can be efficiently combined with the dedicated state-of-the-art dual methods that have recently shown superior results in computer vision and bio-imaging applications. As our empirical evaluation on a wide variety of graph matching datasets suggests, fusion moves significantly improve performance of these methods in terms of speed and quality of the obtained solutions. Our method sets a new state-of-the-art with a notable margin with respect to its competitors.
Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields such as computer vision and biology. Recently, there has been a plethora of work studying efficient algorithms for graph matching under probabilistic models. In this work, we propose a new algorithm for graph matching: Our algorithm associates each vertex with a signature vector using a multistage procedure and then matches a pair of vertices from the two graphs if their signature vectors are close to each other. We show that, for two Erd\H{o}s--R\'enyi graphs with edge correlation $1-\alpha$, our algorithm recovers the underlying matching exactly with high probability when $\alpha \le 1 / (\log \log n)^C$, where $n$ is the number of vertices in each graph and $C$ denotes a positive universal constant. This improves the condition $\alpha \le 1 / (\log n)^C$ achieved in previous work.
This thesis is a proof of concept for the potential of Variational Auto-Encoder (VAE) on representation learning of real-world Knowledge Graphs (KG). Inspired by successful approaches to the generation of molecular graphs, we evaluate the capabilities of our model, the Relational Graph Variational Auto-Encoder (RGVAE). The impact of the modular hyperparameter choices, encoding through graph convolutions, graph matching and latent space prior, is compared. The RGVAE is first evaluated on link prediction. The mean reciprocal rank (MRR) scores on the two datasets FB15K-237 and WN18RR are compared to the embedding-based model DistMult. A variational DistMult and a RGVAE without latent space prior constraint are implemented as control models. The results show that between different settings, the RGVAE with relaxed latent space, scores highest on both datasets, yet does not outperform the DistMult. Further, we investigate the latent space in a twofold experiment: first, linear interpolation between the latent representation of two triples, then the exploration of each latent dimension in a $95\%$ confidence interval. Both interpolations show that the RGVAE learns to reconstruct the adjacency matrix but fails to disentangle. For the last experiment we introduce a new validation method for the FB15K-237 data set. The relation type-constrains of generated triples are filtered and matched with entity types. The observed rate of valid generated triples is insignificantly higher than the random threshold. All generated and valid triples are unseen. A comparison between different latent space priors, using the $\delta$-VAE method, reveals a decoder collapse. Finally we analyze the limiting factors of our approach compared to molecule generation and propose solutions for the decoder collapse and successful representation learning of multi-relational KGs.
In this paper we present AMALGAM, a matching approach to fairify tabular data with the use of a knowledge graph. The ultimate goal is to provide fast and efficient approach to annotate tabular data with entities from a background knowledge. The approach combines lookup and filtering services combined with text pre-processing techniques. Experiments conducted in the context of the 2020 Semantic Web Challenge on Tabular Data to Knowledge Graph Matching with both Column Type Annotation and Cell Type Annotation tasks showed promising results.
The graph is one of the most widely used mathematical structures in engineering and science because of its representational power and inherent ability to demonstrate the relationship between objects. The objective of this work is to introduce the novel graph matching techniques using the representational power of the graph and apply it to structural pattern recognition applications. We present an extensive survey of various exact and inexact graph matching techniques. Graph matching using the concept of homeomorphism is presented. A category of graph matching algorithms is presented, which reduces the graph size by removing the less important nodes using some measure of relevance. We present an approach to error-tolerant graph matching using node contraction where the given graph is transformed into another graph by contracting smaller degree nodes. We use this scheme to extend the notion of graph edit distance, which can be used as a trade-off between execution time and accuracy. We describe an approach to graph matching by utilizing the various node centrality information, which reduces the graph size by removing a fraction of nodes from both graphs based on a given centrality measure. The graph matching problem is inherently linked to the geometry and topology of graphs. We introduce a novel approach to measure graph similarity using geometric graphs. We define the vertex distance between two geometric graphs using the position of their vertices and show it to be a metric over the set of all graphs with vertices only. We define edge distance between two graphs based on the angular orientation, length and position of the edges. Then we combine the notion of vertex distance and edge distance to define the graph distance between two geometric graphs and show it to be a metric. Finally, we use the proposed graph similarity framework to perform exact and error-tolerant graph matching.
Graph matching (GM) has been a building block in various areas including computer vision and pattern recognition. Despite recent impressive progress, existing deep GM methods often have obvious difficulty in handling outliers, which are ubiquitous in practice. We propose a deep reinforcement learning based approach RGM, whose sequential node matching scheme naturally fits the strategy for selective inlier matching against outliers. A revocable action framework is devised to improve the agent's flexibility against the complex constrained GM. Moreover, we propose a quadratic approximation technique to regularize the affinity score, in the presence of outliers. As such, the agent can finish inlier matching timely when the affinity score stops growing, for which otherwise an additional parameter i.e. the number of inliers is needed to avoid matching outliers. In this paper, we focus on learning the back-end solver under the most general form of GM: the Lawler's QAP, whose input is the affinity matrix. Especially, our approach can also boost existing GM methods that use such input. Experiments on multiple real-world datasets demonstrate its performance regarding both accuracy and robustness.
Matching articulated shapes represented by voxel-sets reduces to maximal sub-graph isomorphism when each set is described by a weighted graph. Spectral graph theory can be used to map these graphs onto lower dimensional spaces and match shapes by aligning their embeddings in virtue of their invariance to change of pose. Classical graph isomorphism schemes relying on the ordering of the eigenvalues to align the eigenspaces fail when handling large data-sets or noisy data. We derive a new formulation that finds the best alignment between two congruent $K$-dimensional sets of points by selecting the best subset of eigenfunctions of the Laplacian matrix. The selection is done by matching eigenfunction signatures built with histograms, and the retained set provides a smart initialization for the alignment problem with a considerable impact on the overall performance. Dense shape matching casted into graph matching reduces then, to point registration of embeddings under orthogonal transformations; the registration is solved using the framework of unsupervised clustering and the EM algorithm. Maximal subset matching of non identical shapes is handled by defining an appropriate outlier class. Experimental results on challenging examples show how the algorithm naturally treats changes of topology, shape variations and different sampling densities.
We present a deep neural network to predict structural similarity between 2D layouts by leveraging Graph Matching Networks (GMN). Our network, coined LayoutGMN, learns the layout metric via neural graph matching, using an attention-based GMN designed under a triplet network setting. To train our network, we utilize weak labels obtained by pixel-wise Intersection-over-Union (IoUs) to define the triplet loss. Importantly, LayoutGMN is built with a structural bias which can effectively compensate for the lack of structure awareness in IoUs. We demonstrate this on two prominent forms of layouts, viz., floorplans and UI designs, via retrieval experiments on large-scale datasets. In particular, retrieval results by our network better match human judgement of structural layout similarity compared to both IoUs and other baselines including a state-of-the-art method based on graph neural networks and image convolution. In addition, LayoutGMN is the first deep model to offer both metric learning of structural layout similarity and structural matching between layout elements.
We study the problem of inter cell interference coordination (ICIC) with fixed transmit power in OFDMA-based cellular networks, in which each base station (BS) needs to decide as to which subchannel, if any, to allocate to each of its associated mobile stations (MS) for data transmission. In general, there exists a trade-off between the total throughput (sum of throughputs of all the MSs) and fairness under the allocations found by resource allocation schemes. We introduce the concept of $\tau-\alpha-$fairness by modifying the concept of $\alpha-$fairness, which was earlier proposed in the context of designing fair end-to-end window-based congestion control protocols for packet-switched networks. The concept of $\tau-\alpha-$fairness allows us to achieve arbitrary trade-offs between the total throughput and degree of fairness by selecting an appropriate value of $\alpha$ in $[0,\infty)$. We show that for every $\alpha \in [0,\infty)$ and every $\tau > 0$, the problem of finding a $\tau-\alpha-$fair allocation is NP-Complete. Further, we show that for every $\alpha \in [0, \infty)$, there exist thresholds such that if the potential interference levels experienced by each MS on every subchannel are above the threshold values, then the problem can be optimally solved in polynomial time by reducing it to the bipartite graph matching problem. Also, we propose a simple, distributed subchannel allocation algorithm for the ICIC problem, which is flexible, requires a small amount of time to operate, and requires information exchange among only neighboring BSs. We investigate via simulations as to how the algorithm parameters should be selected so as to achieve any desired trade-off between the total throughput and fairness.
In V2V Mode-3, eNodeBs assign subchannels to vehicles in order for them to periodically broadcast CAM messages \cite{b2}. A crucial aspect is to ensure that vehicles in the same cluster will broadcast in orthogonal time subchannels\footnote{A subchannel is a time-frequency resource chunk capable of sufficiently conveying a CAM message.} to avoid conflicts. In general, resource/subchannel allocation problems can be represented as weighted bipartite graphs. However, in this scenario there is an additional time orthogonality constraint which cannot be straightforwardly handled by conventional graph matching methods \cite{b3}. Thus, in our approach the mentioned constraint has been taken into account. We also perform the allocation task in a sequential manner based on the constrainedness of each cluster. To illustrate the gist of the problem, in Fig. 1 we show two partially overlapping clusters where a conflict between vehicles $V_8$ and $V_{10}$ is generated as the allotted subchannels are in the same subframe.
The problem of learning a correspondence relationship between nodes of two networks has drawn much attention of the computer science community and recently that of statisticians. The unseeded version of this problem, in which we do not know any part of the true correspondence, is a long-standing challenge. For low-rank networks, the problem can be translated into an unsupervised point registration problem, in which two point sets generated from the same distribution are matchable by an unknown orthonormal transformation. Conventional methods generally lack consistency guarantee and are usually computationally costly. In this paper, we propose a novel approach to this problem. Instead of simultaneously estimating the unknown correspondence and orthonormal transformation to match up the two point sets, we match their distributions via minimizing our designed loss function capturing the discrepancy between their Laplace transforms, thus avoiding the optimization over all possible correspondences. This dramatically reduces the dimension of the optimization problem from $\Omega(n^2)$ parameters to $O(d^2)$ parameters, where $d$ is the fixed rank, and enables convenient theoretical analysis. In this paper, we provide arguably the first consistency guarantee and explicit error rate for general low-rank models. Our method provides control over the computational complexity ranging from $\omega(n)$ (any growth rate faster than $n$) to $O(n^2)$ while pertaining consistency. We demonstrate the effectiveness of our method through several numerical examples.
The 3rd Generation Partnership Project (3GPP) has introduced in Rel. 14 a novel technology referred to as vehicle--to--vehicle (V2V) \textit{mode-3}. Under this scheme, the eNodeB assists in the resource allocation process allotting sidelink subchannels to vehicles. Thereupon, vehicles transmit their signals in a broadcast manner without the intervention of the former one. eNodeBs will thereby play a determinative role in the assignment of subchannels as they can effectively manage V2V traffic and prevent allocation conflicts. The latter is a crucial aspect to be enforced in order for the signals to be received reliably by other vehicles. To this purpose, we propose two resource allocation schemes namely bipartite graph matching-based successive allocation (BGM-SA) and bipartite graph matching-based parallel allocation (BGM-PA) which are suboptimal approaches with lesser complexity than exhaustive search. Both schemes incorporate constraints to prevent allocation conflicts from emerging. In this research, we consider overlapping clusters only, which could be formed at intersections or merging highways. We show through simulations that BGM-SA can attain near-optimal performance whereas BGM-PA is subpar but less complex. Additionally, since BGM-PA is based on inter-cluster vehicle pre-grouping, we explore different metrics that could effectively portray the overall channel conditions of pre-grouped vehicles. This is of course not optimal in terms of maximizing the system capacity---since the allocation process would be based on simplified surrogate information---but it reduces the computational complexity.
In this paper we present a graph-based resource allocation scheme for sidelink broadcast vehicle-to-vehicle (V2V) communications. Harnessing available information on the geographical position of vehicles and spectrum resources utilization, eNodeBs are capable of allotting the same set of sidelink resources to several different vehicles in order for them to broadcast their signals. Hence, vehicles sharing the same resources would ideally be in different communications clusters for the interference level-generated due to resource repurposing-to be maintained under control. Within a communications cluster, it is crucial that vehicles transmit in orthogonal time resources to prevent conflicts as vehicles-with half-duplex radio interfaces--cannot transmit and receive simultaneously. In this research, we have envisaged a solution based on a bipartite graph, where vehicles and spectrum resources are represented by vertices whereas the edges represent the achievable rate in each resource based on the signal-to-interference-plus-noise ratio (SINR) that vehicles perceive. The aforementioned constraint on time orthogonality of allocated resources can be approached by aggregating conflicting vertices into macro-vertices which, in addition, narrows the search space yielding a solution with computational complexity equivalent to the conventional graph matching problem. We show mathematically and through simulations that the proposed approach yields an optimal solution. In addition, we provide simulations showing that the proposed method outperforms other competing approaches, specially in scenarios with high vehicular density.
Graph matching finds the correspondence of nodes across two graphs and is a basic task in graph-based machine learning. Numerous existing methods match every node in one graph to one node in the other graph whereas two graphs usually overlap partially in many \realworld{} applications. In this paper, a partial Gromov-Wasserstein learning framework is proposed for partially matching two graphs, which fuses the partial Gromov-Wasserstein distance and the partial Wasserstein distance as the objective and updates the partial transport map and the node embedding in an alternating fashion. The proposed framework transports a fraction of the probability mass and matches node pairs with high relative similarities across the two graphs. Incorporating an embedding learning method, heterogeneous graphs can also be matched. Numerical experiments on both synthetic and \realworld{} graphs demonstrate that our framework can improve the F1 score by at least $20\%$ and often much more.
Correlative microscopy aims at combining two or more modalities to gain more information than the one provided by one modality on the same biological structure. Registration is needed at different steps of correlative microscopies workflows. Biologists want to select the image content used for registration not to introduce bias in the correlation of unknown structures. Intensity-based methods might not allow this selection and might be too slow when the images are very large. We propose an approach based on point clouds created from selected content by the biologist. These point clouds may be prone to big differences in densities but also missing parts and outliers. In this paper we present a method of registration for point clouds based on graph building and graph matching, and compare the method to iterative closest point based methods.
Grounding language queries in videos aims at identifying the time interval (or moment) semantically relevant to a language query. The solution to this challenging task demands understanding videos' and queries' semantic content and the fine-grained reasoning about their multi-modal interactions. Our key idea is to recast this challenge into an algorithmic graph matching problem. Fueled by recent advances in Graph Neural Networks, we propose to leverage Graph Convolutional Networks to model video and textual information as well as their semantic alignment. To enable the mutual exchange of information across the modalities, we design a novel Video-Language Graph Matching Network (VLG-Net) to match video and query graphs. Core ingredients include representation graphs built atop video snippets and query tokens separately and used to model intra-modality relationships. A Graph Matching layer is adopted for cross-modal context modeling and multi-modal fusion. Finally, moment candidates are created using masked moment attention pooling by fusing the moment's enriched snippet features. We demonstrate superior performance over state-of-the-art grounding methods on three widely used datasets for temporal localization of moments in videos with language queries: ActivityNet-Captions, TACoS, and DiDeMo.
Tensor Kronecker products, the natural generalization of the matrix Kronecker product, are independently emerging in multiple research communities. Like their matrix counterpart, the tensor generalization gives structure for implicit multiplication and factorization theorems. We present a theorem that decouples the dominant eigenvectors of tensor Kronecker products, which is a rare generalization from matrix theory to tensor eigenvectors. This theorem implies low rank structure ought to be present in the iterates of tensor power methods on Kronecker products. We investigate low rank structure in the network alignment algorithm TAME, a power method heuristic. Using the low rank structure directly or via a new heuristic embedding approach, we produce new algorithms which are faster while improving or maintaining accuracy, and scale to problems that cannot be realistically handled with existing techniques.
Code retrieval is to find the code snippet from a large corpus of source code repositories that highly matches the query of natural language description. Recent work mainly uses natural language processing techniques to process both query texts (i.e., human natural language) and code snippets (i.e., machine programming language), however neglecting the deep structured features of query texts and source codes, both of which contain rich semantic information. In this paper, we propose an end-to-end deep graph matching and searching (DGMS) model based on graph neural networks for the task of semantic code retrieval. To this end, we first represent both natural language query texts and programming language code snippets with the unified graph-structured data, and then use the proposed graph matching and searching model to retrieve the best matching code snippet. In particular, DGMS not only captures more structural information for individual query texts or code snippets but also learns the fine-grained similarity between them by cross-attention based semantic matching operations. We evaluate the proposed DGMS model on two public code retrieval datasets with two representative programming languages (i.e., Java and Python). Experiment results demonstrate that DGMS significantly outperforms state-of-the-art baseline models by a large margin on both datasets. Moreover, our extensive ablation studies systematically investigate and illustrate the impact of each part of DGMS.
The core problem of visual multi-robot simultaneous localization and mapping (MR-SLAM) is how to efficiently and accurately perform multi-robot global localization (MR-GL). The difficulties are two-fold. The first is the difficulty of global localization for significant viewpoint difference. Appearance-based localization methods tend to fail under large viewpoint changes. Recently, semantic graphs have been utilized to overcome the viewpoint variation problem. However, the methods are highly time-consuming, especially in large-scale environments. This leads to the second difficulty, which is how to perform real-time global localization. In this paper, we propose a semantic histogram-based graph matching method that is robust to viewpoint variation and can achieve real-time global localization. Based on that, we develop a system that can accurately and efficiently perform MR-GL for both homogeneous and heterogeneous robots. The experimental results show that our approach is about 30 times faster than Random Walk based semantic descriptors. Moreover, it achieves an accuracy of 95% for global localization, while the accuracy of the state-of-the-art method is 85%.