On Multi-Relational Link Prediction with Bilinear Models

We study bilinear embedding models for the task of multi-relational link prediction and knowledge graph completion. Bilinear models belong to the most basic models for this task, they are comparably efficient to train and use, and they can provide good prediction performance. The main goal of this paper is to explore the expressiveness of and the connections between various bilinear models proposed in the literature. In particular, a substantial number of models can be represented as bilinear models with certain additional constraints enforced on the embeddings. We explore whether or not these constraints lead to universal models, which can in principle represent every set of relations, and whether or not there are subsumption relationships between various models. We report results of an independent experimental study that evaluates recent bilinear models in a common experimental setup. Finally, we provide evidence that relation-level ensembles of multiple bilinear models can achieve state-of-the art prediction performance

Large Margin Nearest Neighbor Embedding for Knowledge Representation

Traditional way of storing facts in triplets ({\it head\_entity, relation, tail\_entity}), abbreviated as ({\it h, r, t}), makes the knowledge intuitively displayed and easily acquired by mankind, but hardly computed or even reasoned by AI machines. Inspired by the success in applying {\it Distributed Representations} to AI-related fields, recent studies expect to represent each entity and relation with a unique low-dimensional embedding, which is different from the symbolic and atomic framework of displaying knowledge in triplets. In this way, the knowledge computing and reasoning can be essentially facilitated by means of a simple {\it vector calculation}, i.e. ${\bf h} + {\bf r} \approx {\bf t}$. We thus contribute an effective model to learn better embeddings satisfying the formula by pulling the positive tail entities ${\bf t^{+}}$ to get together and close to {\bf h} + {\bf r} ({\it Nearest Neighbor}), and simultaneously pushing the negatives ${\bf t^{-}}$ away from the positives ${\bf t^{+}}$ via keeping a {\it Large Margin}. We also design a corresponding learning algorithm to efficiently find the optimal solution based on {\it Stochastic Gradient Descent} in iterative fashion. Quantitative experiments illustrate that our approach can achieve the state-of-the-art performance, compared with several latest methods on some benchmark datasets for two classical applications, i.e. {\it Link prediction} and {\it Triplet classification}. Moreover, we analyze the parameter complexities among all the evaluated models, and analytical results indicate that our model needs fewer computational resources on outperforming the other methods.Comment: arXiv admin note: text overlap with arXiv:1503.0815

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page