We propose learning graph representations from 2D feature maps for visual recognition. Our method draws inspiration from region based recognition, and learns to transform a 2D image into a graph structure. The vertices of the graph define clusters of pixels (regions), and the edges measure the similarity between these clusters in a feature space. Our method further learns to propagate information across all vertices on the graph, and is able to project the learned graph representation back into 2D grids. Our graph representation facilitates reasoning beyond regular grids and can capture long range dependencies among regions. We demonstrate that our model can be trained from end-to-end, and is easily integrated into existing networks. Finally, we evaluate our method on three challenging recognition tasks: semantic segmentation, object detection and object instance segmentation. For all tasks, our method outperforms state-of-the-art methods.

Most real-world networks are too large to be measured or studied directly and there is substantial interest in estimating global network properties from smaller sub-samples. One of the most important global properties is the number of vertices/nodes in the network. Estimating the number of vertices in a large network is a major challenge in computer science, epidemiology, demography, and intelligence analysis. In this paper we consider a population random graph G = (V;E) from the stochastic block model (SBM) with K communities/blocks. A sample is obtained by randomly choosing a subset W and letting G(W) be the induced subgraph in G of the vertices in W. In addition to G(W), we observe the total degree of each sampled vertex and its block membership. Given this partial information, we propose an efficient PopULation Size Estimation algorithm, called PULSE, that accurately estimates the size of the whole population as well as the size of each community. To support our theoretical analysis, we perform an exhaustive set of experiments to study the effects of sample size, K, and SBM model parameters on the accuracy of the estimates. The experimental results also demonstrate that PULSE significantly outperforms a widely-used method called the network scale-up estimator in a wide variety of scenarios.

Many popular network models rely on the assumption of (vertex) exchangeability, in which the distribution of the graph is invariant to relabelings of the vertices. However, the Aldous-Hoover theorem guarantees that these graphs are dense or empty with probability one, whereas many real-world graphs are sparse. We present an alternative notion of exchangeability for random graphs, which we call edge exchangeability, in which the distribution of a graph sequence is invariant to the order of the edges. We demonstrate that edge-exchangeable models, unlike models that are traditionally vertex exchangeable, can exhibit sparsity. To do so, we outline a general framework for graph generative models; by contrast to the pioneering work of Caron and Fox (2015), models within our framework are stationary across steps of the graph sequence. In particular, our model grows the graph by instantiating more latent atoms of a single random measure as the dataset size increases, rather than adding new atoms to the measure.

We present a learning-based approach to computing solutions for certain NP-hard problems. Our approach combines deep learning techniques with useful algorithmic elements from classic heuristics. The central component is a graph convolutional network that is trained to estimate the likelihood, for each vertex in a graph, of whether this vertex is part of the optimal solution. The network is designed and trained to synthesize a diverse set of solutions, which enables rapid exploration of the solution space via tree search. The presented approach is evaluated on four canonical NP-hard problems and five datasets, which include benchmark satisfiability problems and real social network graphs with up to a hundred thousand nodes. Experimental results demonstrate that the presented approach substantially outperforms recent deep learning work, and performs on par with highly optimized state-of-the-art heuristic solvers for some NP-hard problems. Experiments indicate that our approach generalizes across datasets, and scales to graphs that are orders of magnitude larger than those used during training.

In this paper we address the problem of Maximum Inner Product Search (MIPS) that is currently the computational bottleneck in a large number of machine learning applications. While being similar to the nearest neighbor search (NNS), the MIPS problem was shown to be more challenging, as the inner product is not a proper metric function. We propose to solve the MIPS problem with the usage of similarity graphs, i.e., graphs where each vertex is connected to the vertices that are the most similar in terms of some similarity function. Originally, the framework of similarity graphs was proposed for metric spaces and in this paper we naturally extend it to the non-metric MIPS scenario. We demonstrate that, unlike existing approaches, similarity graphs do not require any data transformation to reduce MIPS to the NNS problem and should be used for the original data. Moreover, we explain why such a reduction is detrimental for similarity graphs. By an extensive comparison to the existing approaches, we show that the proposed method is a game-changer in terms of the runtime/accuracy trade-off for the MIPS problem.

Stochastic network models play a central role across a wide range of scientific disciplines, and questions of statistical inference arise naturally in this context. In this paper we investigate goodness-of-fit and two-sample testing procedures for statistical networks based on the principle of maximum entropy (MaxEnt). Our approach formulates a constrained entropy-maximization problem on the space of networks, subject to prescribed structural constraints. The resulting test statistics are defined through the Lagrange multipliers associated with the constrained optimization problem, which, to our knowledge, is novel in the statistical networks literature. We establish consistency in the classical regime where the number of vertices is fixed. We then consider asymptotic regimes in which the graph size grows with the sample size, developing tests for both dense and sparse settings. In the dense case, we analyze exponential random graph models (ERGM) (including the Erdös-Rènyi models), while in the sparse regime our theory applies to Erd{ö}s-R{è}nyi graphs. Our analysis leverages recent advances in nonlinear large deviation theory for random graphs. We further show that the proposed Lagrange-multiplier framework connects naturally to classical score tests for constrained maximum likelihood estimation. The results provide a unified entropy-based framework for network model assessment across diverse growth regimes.

Neural Information Processing Systems http://nips.cc/

Hypergraphs are important objects to model ternary or higher-order relations of objects, and haveanumber ofapplications inanalysing manycomplexdatasets occurring in practice.