We propose a deep learning-based approach to the problem of premise selection: selecting mathematical statements relevant for proving a given conjecture. We represent a higher-order logic formula as a graph that is invariant to variable renaming but still fully preserves syntactic and semantic information. We then embed the graph into a vector via a novel embedding method that preserves the information of edge ordering. Our approach achieves state-of-the-art results on the HolStep dataset, improving the classification accuracy from 83 to 90.3 .
Our graph embedding method is related to a large body of prior work on embeddings and graphs. Deepwalk @cite_3 , LINE @cite_52 and Node2Vec @cite_23 focus on learning node embeddings. Similar to Word2Vec @cite_19 @cite_42 , they optimize the embedding of a node to predict nodes in a neighborhood. Recursive neural networks @cite_2 @cite_24 and Tree LSTMs @cite_15 consider embeddings of trees, a special type of graphs. Neural networks on general graphs were first introduced by @cite_43 and @cite_10 . Many follow-up works @cite_13 @cite_27 @cite_6 @cite_29 @cite_17 @cite_37 proposed specific architectures to handle graph-based input by extending recurrent neural network to graph data @cite_43 @cite_35 @cite_21 or making use of graph convolutions based on spectral graph theories @cite_13 @cite_29 @cite_53 @cite_37 @cite_30 . Our approach is most similar to the work of @cite_13 , where they encode molecular fragments as neural fingerprints with graph-based convolutions for chemical applications. But to the best of our knowledge, no previous deep learning approaches on general graphs preserve the order of edges. In contrast, we propose a novel way of graph embedding that can preserve the information of edge ordering, and demonstrate its effectiveness for premise selection.
@cite_30 @cite_35 @cite_29 @cite_42 @cite_3 @cite_43 @cite_2 @cite_15 @cite_10 @cite_21 @cite_52 @cite_23 @cite_17 @cite_37 @cite_6 @cite_19 @cite_27 @cite_53 @cite_24 @cite_13
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We propose a new method for embedding graphs while preserving directed edge information. Learning such continuous-space vector representations (or embeddings) of nodes in a graph is an important first step for using network information (from social networks, user-item graphs, knowledge bases, etc.) in many machine learning tasks. Unlike previous work, we (1) explicitly model an edge as a function of node embeddings, and we (2) propose a novel objective, the graph likelihood, which contrasts information from sampled random walks with non-existent edges. Individually, both of these contributions improve the learned representations, especially when there are memory constraints on the total size of the embeddings. When combined, our contributions enable us to significantly improve the state-of-the-art by learning more concise representations that better preserve the graph structure. We evaluate our method on a variety of link-prediction task including social networks, collaboration networks, and protein interactions, showing that our proposed method learn representations with error reductions of up to 76 and 55 , on directed and undirected graphs. In addition, we show that the representations learned by our method are quite space efficient, producing embeddings which have higher structure-preserving accuracy but are 10 times smaller. Recent advancements in deep neural networks for graph-structured data have led to state-of-the-art performance on recommender system benchmarks. However, making these methods practical and scalable to web-scale recommendation tasks with billions of items and hundreds of millions of users remains an unsolved challenge. Here we describe a large-scale deep recommendation engine that we developed and deployed at Pinterest. We develop a data-efficient Graph Convolutional Network (GCN) algorithm, which combines efficient random walks and graph convolutions to generate embeddings of nodes (i.e., items) that incorporate both graph structure as well as node feature information. Compared to prior GCN approaches, we develop a novel method based on highly efficient random walks to structure the convolutions and design a novel training strategy that relies on harder-and-harder training examples to improve robustness and convergence of the model. We also develop an efficient MapReduce model inference algorithm to generate embeddings using a trained model. Overall, we can train on and embed graphs that are four orders of magnitude larger than typical GCN implementations. We show how GCN embeddings can be used to make high-quality recommendations in various settings at Pinterest, which has a massive underlying graph with 3 billion nodes representing pins and boards, and 17 billion edges. According to offline metrics, user studies, as well as A B tests, our approach generates higher-quality recommendations than comparable deep learning based systems. To our knowledge, this is by far the largest application of deep graph embeddings to date and paves the way for a new generation of web-scale recommender systems based on graph convolutional architectures. Celebrated and its fruitful variants are powerful models to achieve excellent performance on the tasks that map sequences to sequences. However, these are many machine learning tasks with inputs naturally represented in a form of graphs, which imposes significant challenges to existing Seq2Seq models for lossless conversion from its graph form to the sequence. In this work, we present a general end-to-end approach to map the input graph to a sequence of vectors, and then another attention-based LSTM to decode the target sequence from these vectors. Specifically, to address inevitable information loss for data conversion, we introduce a novel graph-to-sequence neural network model that follows the encoder-decoder architecture. Our method first uses an improved graph-based neural network to generate the node and graph embeddings by a novel aggregation strategy to incorporate the edge direction information into the node embeddings. We also propose an attention based mechanism that aligns node embeddings and decoding sequence to better cope with large graphs. Experimental results on bAbI task, Shortest Path Task, and Natural Language Generation Task demonstrate that our model achieves the state-of-the-art performance and significantly outperforms other baselines. We also show that with the proposed aggregation strategy, our proposed model is able to quickly converge to good performance. Network embedding methodologies, which learn a distributed vector representation for each vertex in a network, have attracted considerable interest in recent years. Existing works have demonstrated that vertex representation learned through an embedding method provides superior performance in many real-world applications, such as node classification, link prediction, and community detection. However, most of the existing methods for network embedding only utilize topological information of a vertex, ignoring a rich set of nodal attributes (such as, user profiles of an online social network, or textual contents of a citation network), which is abundant in all real-life networks. A joint network embedding that takes into account both attributional and relational information entails a complete network information and could further enrich the learned vector representations. In this work, we present Neural-Brane, a novel Neural Bayesian Personalized Ranking based Attributed Network Embedding. For a given network, Neural-Brane extracts latent feature representation of its vertices using a designed neural network model that unifies network topological information and nodal attributes; Besides, it utilizes Bayesian personalized ranking objective, which exploits the proximity ordering between a similar node-pair and a dissimilar node-pair. We evaluate the quality of vertex embedding produced by Neural-Brane by solving the node classification and clustering tasks on four real-world datasets. Experimental results demonstrate the superiority of our proposed method over the state-of-the-art existing methods.