This paper tackles the problem of endogenous link prediction for Knowledge Base completion. Knowledge Bases can be represented as directed graphs whose nodes correspond to entities and edges to relationships. Previous attempts either consist of powerful systems with high capacity to model complex connectivity patterns, which unfortunately usually end up overfitting on rare relationships, or in approaches that trade capacity for simplicity in order to fairly model all relationships, frequent or not. In this paper, we propose Tatec a happy medium obtained by complementing a high-capacity model with a simpler one, both pre-trained separately and then combined. We present several variants of this model with different kinds of regularization and combination strategies and show that this approach outperforms existing methods on different types of relationships by achieving state-of-the-art results on four benchmarks of the literature.
One of the simplest and most successful 2-way models is @cite_15 . In that model, relationships are represented as translations in the embedding space: if @math holds, then the embedding of the @math should be close to the embedding of @math plus some vector that depends on the @math . This is a natural approach to model hierarchical and asymmetric relationships, which are common in knowledge bases such as Freebase . Several modifications to have been proposed recently, @cite_8 and @cite_25 . In , the embeddings of the entities are projected onto a hyperplane that depends on @math before the translation. The second algorithm, , follows the same idea, except that the projection operator is a matrix that is more general than an orthogonal projection to a hyperplane. As we shall see in the next section, corresponds to our model with additional constraints on the parameters.
@cite_15 @cite_25 @cite_8
2127795553 2184957013 2283196293
This paper proposes a novel translation-based knowledge graph embedding that preserves the logical properties of relations such as transitivity and symmetricity. The embedding space generated by existing translation-based embeddings do not represent transitive and symmetric relations precisely, because they ignore the role of entities in triples. Thus, we introduce a role-specific projection which maps an entity to distinct vectors according to its role in a triple. That is, a head entity is projected onto an embedding space by a head projection operator, and a tail entity is projected by a tail projection operator. This idea is applied to TransE, TransR, and TransD to produce lppTransE, lppTransR, and lppTransD, respectively. According to the experimental results on link prediction and triple classification, the proposed logical property preserving embeddings show the state-of-the-art performance at both tasks. These results prove that it is critical to preserve logical properties of relations while embedding knowledge graphs, and the proposed method does it effectively. This paper considers the problem of knowledge inference on large-scale imperfect repositories with incomplete coverage by means of embedding entities and relations at the first attempt. We propose IIKE (Imperfect and Incomplete Knowledge Embedding), a probabilistic model which measures the probability of each belief, i.e. @math , in large-scale knowledge bases such as NELL and Freebase, and our objective is to learn a better low-dimensional vector representation for each entity ( @math and @math ) and relation ( @math ) in the process of minimizing the loss of fitting the corresponding confidence given by machine learning (NELL) or crowdsouring (Freebase), so that we can use @math to assess the plausibility of a belief when conducting inference. We use subsets of those inexact knowledge bases to train our model and test the performances of link prediction and triplet classification on ground truth beliefs, respectively. The results of extensive experiments show that IIKE achieves significant improvement compared with the baseline and state-of-the-art approaches. For any integers @math and @math , we show the existence of a set of @math vectors @math such that any embedding @math satisfying @math must have @math This lower bound matches the upper bound given by the Johnson-Lindenstrauss lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of @math of interest, since there is always an isometric embedding into dimension @math (either the identity map, or projection onto @math ). Previously such a lower bound was only known to hold against linear maps @math , and not for such a wide range of parameters @math [LN16]. The best previously known lower bound for general @math was @math [Wel74, Lev83, Alo03], which is suboptimal for any @math . We introduce a new low-distortion embedding of @math into @math ( @math ) called the fast Johnson-Lindenstrauss transform (FJLT). The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a randomized Fourier transform. Sparse random projections are unsuitable for low-distortion embeddings. We overcome this handicap by exploiting the “Heisenberg principle” of the Fourier transform, i.e., its local-global duality. The FJLT can be used to speed up search algorithms based on low-distortion embeddings in @math and @math . We consider the case of approximate nearest neighbors in @math . We provide a faster algorithm using classical projections, which we then speed up further by plugging in the FJLT. We also give a faster algorithm for searching over the hypercube. Binary embedding of high-dimensional data requires long codes to preserve the discriminative power of the input space. Traditional binary coding methods often suffer from very high computation and storage costs in such a scenario. To address this problem, we propose Circulant Binary Embedding (CBE) which generates binary codes by projecting the data with a circulant matrix. The circulant structure enables the use of Fast Fourier Transformation to speed up the computation. Compared to methods that use unstructured matrices, the proposed method improves the time complexity from @math to @math , and the space complexity from @math to @math where @math is the input dimensionality. We also propose a novel time-frequency alternating optimization to learn data-dependent circulant projections, which alternatively minimizes the objective in original and Fourier domains. We show by extensive experiments that the proposed approach gives much better performance than the state-of-the-art approaches for fixed time, and provides much faster computation with no performance degradation for fixed number of bits. Knowledge graph completion aims to perform link prediction between entities. In this paper, we consider the approach of knowledge graph embeddings. Recently, models such as TransE and TransH build entity and relation embeddings by regarding a relation as translation from head entity to tail entity. We note that these models simply put both entities and relations within the same semantic space. In fact, an entity may have multiple aspects and various relations may focus on different aspects of entities, which makes a common space insufficient for modeling. In this paper, we propose TransR to build entity and relation embeddings in separate entity space and relation spaces. Afterwards, we learn embeddings by first projecting entities from entity space to corresponding relation space and then building translations between projected entities. In experiments, we evaluate our models on three tasks including link prediction, triple classification and relational fact extraction. Experimental results show significant and consistent improvements compared to state-of-the-art baselines including TransE and TransH. The source code of this paper can be obtained from https: github.com mrlyk423 relation_extraction. Distributions over permutations arise in applications ranging from multi-object tracking to ranking of instances. The difficulty of dealing with these distributions is caused by the size of their domain, which is factorial in the number of considered entities ( @math ). It makes the direct definition of a multinomial distribution over permutation space impractical for all but a very small @math . In this work we propose an embedding of all @math permutations for a given @math in a surface of a hypersphere defined in @math . As a result of the embedding, we acquire ability to define continuous distributions over a hypersphere with all the benefits of directional statistics. We provide polynomial time projections between the continuous hypersphere representation and the @math -element permutation space. The framework provides a way to use continuous directional probability densities and the methods developed thereof for establishing densities over permutations. As a demonstration of the benefits of the framework we derive an inference procedure for a state-space model over permutations. We demonstrate the approach with applications. Recurrent neural networks (RNNs) have achieved state-of-the-art performances in many natural language processing tasks, such as language modeling and machine translation. However, when the vocabulary is large, the RNN model will become very big (e.g., possibly beyond the memory capacity of a GPU device) and its training will become very inefficient. In this work, we propose a novel technique to tackle this challenge. The key idea is to use 2-Component (2C) shared embedding for word representations. We allocate every word in the vocabulary into a table, each row of which is associated with a vector, and each column associated with another vector. Depending on its position in the table, a word is jointly represented by two components: a row vector and a column vector. Since the words in the same row share the row vector and the words in the same column share the column vector, we only need @math vectors to represent a vocabulary of @math unique words, which are far less than the @math vectors required by existing approaches. Based on the 2-Component shared embedding, we design a new RNN algorithm and evaluate it using the language modeling task on several benchmark datasets. The results show that our algorithm significantly reduces the model size and speeds up the training process, without sacrifice of accuracy (it achieves similar, if not better, perplexity as compared to state-of-the-art language models). Remarkably, on the One-Billion-Word benchmark Dataset, our algorithm achieves comparable perplexity to previous language models, whilst reducing the model size by a factor of 40-100, and speeding up the training process by a factor of 2. We name our proposed algorithm to reflect its very small model size and very high training speed. We give efficient algorithms for volume sampling, i.e., for picking @math -subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). In other words, we can efficiently sample @math -subsets of @math with probabilities proportional to the corresponding @math by @math principal minors of any given @math by @math positive semi definite matrix. This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section @math of KV , also implicit in BDM, DRVW ). Our first algorithm for volume sampling @math -subsets of rows from an @math -by- @math matrix runs in @math arithmetic operations (where @math is the exponent of matrix multiplication) and a second variant of it for @math -approximate volume sampling runs in @math arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small @math . Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: (1) Given @math , in @math arithmetic operations we can find @math of its rows such that projecting onto their span gives a @math -approximation to the matrix of rank @math closest to @math under the Frobenius norm. This improves the @math -approximation of Boutsidis, Drineas and Mahoney BDM and matches the lower bound shown in DRVW . The method of conditional expectations gives a algorithm with the same complexity. The running time can be improved to @math at the cost of losing an extra @math in the approximation factor. (2) The same rows and projection as in the previous point give a @math -approximation to the matrix of rank @math closest to @math under the spectral norm. In this paper, we show an almost matching lower bound of @math , even for @math . We prove an incidence theorem for points and planes in the projective space @math over any field @math , whose characteristic @math An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving @math if @math . This yields a bound on the number of incidences between @math points and @math planes in @math , with @math as @math where @math is the maximum number of collinear planes, provided that @math if @math . Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when @math . For a non-collinear point set @math and a non-degenerate symmetric or skew-symmetric bilinear form @math , the number of distinct values of @math on pairs of points of @math is @math . This is also the best known bound over @math , where it follows from the Szemer 'edi-Trotter theorem. Also, a set @math , not supported in a single semi-isotropic plane contains a point, from which @math distinct distances to other points of @math are attained. Knowledge representation is a major topic in AI, and many studies attempt to represent entities and relations of knowledge base in a continuous vector space. Among these attempts, translation-based methods build entity and relation vectors by minimizing the translation loss from a head entity to a tail one. In spite of the success of these methods, translation-based methods also suffer from the oversimplified loss metric, and are not competitive enough to model various and complex entities relations in knowledge bases. To address this issue, we propose , an adaptive metric approach for embedding, utilizing the metric learning ideas to provide a more flexible embedding method. Experiments are conducted on the benchmark datasets and our proposed method makes significant and consistent improvements over the state-of-the-art baselines. We deal with embedding a large scale knowledge graph composed of entities and relations into a continuous vector space. TransE is a promising method proposed recently, which is very efficient while achieving state-of-the-art predictive performance. We discuss some mapping properties of relations which should be considered in embedding, such as reflexive, one-to-many, many-to-one, and many-to-many. We note that TransE does not do well in dealing with these properties. Some complex models are capable of preserving these mapping properties but sacrifice efficiency in the process. To make a good trade-off between model capacity and efficiency, in this paper we propose TransH which models a relation as a hyperplane together with a translation operation on it. In this way, we can well preserve the above mapping properties of relations with almost the same model complexity of TransE. Additionally, as a practical knowledge graph is often far from completed, how to construct negative examples to reduce false negative labels in training is very important. Utilizing the one-to-many many-to-one mapping property of a relation, we propose a simple trick to reduce the possibility of false negative labeling. We conduct extensive experiments on link prediction, triplet classification and fact extraction on benchmark datasets like WordNet and Freebase. Experiments show TransH delivers significant improvements over TransE on predictive accuracy with comparable capability to scale up. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We consider the problem of embedding entities and relationships of multi-relational data in low-dimensional vector spaces. Our objective is to propose a canonical model which is easy to train, contains a reduced number of parameters and can scale up to very large databases. Hence, we propose TransE, a method which models relationships by interpreting them as translations operating on the low-dimensional embeddings of the entities. Despite its simplicity, this assumption proves to be powerful since extensive experiments show that TransE significantly outperforms state-of-the-art methods in link prediction on two knowledge bases. Besides, it can be successfully trained on a large scale data set with 1M entities, 25k relationships and more than 17M training samples. We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any @math , given cost and demand graphs with edge weights @math respectively, we can find a set @math with @math at most @math times the optimal non-uniform sparsest cut value, in time @math provided @math . Here @math is the @math 'th smallest generalized eigenvalue of the Laplacian matrices of cost and demand graphs; @math (resp. @math ) is the weight of edges crossing the @math cut in cost (resp. demand) graph and @math is the sparsity of the optimal cut. In words, we show that the non-uniform sparsest cut problem is easy when the generalized spectrum grows moderately fast. To the best of our knowledge, there were no results based on higher order spectra for non-uniform sparsest cut prior to this work. Even for uniform sparsest cut, the quantitative aspects of our result are somewhat stronger than previous methods. Similar results hold for other expansion measures like edge expansion, normalized cut, and conductance, with the @math 'th smallest eigenvalue of the normalized Laplacian playing the role of @math in the latter two cases. Our proof is based on an l1-embedding of vectors from a semi-definite program from the Lasserre hierarchy. The embedded vectors are then rounded to a cut using standard threshold rounding. We hope that the ideas connecting @math -embeddings to Lasserre SDPs will find other applications. Another aspect of the analysis is the adaptation of the column selection paradigm from our earlier work on rounding Lasserre SDPs [GS11] to pick a set of edges rather than vertices. This feature is important in order to extend the algorithms to non-uniform sparsest cut. The set maxima problem is as follows: given a family of subsets @math of a totally ordered set @math , find the maximum in each subset. The computational model is the comparison tree. One possible solution is to sort the set X, which requires @math comparisons. The open question is whether set maxima is easier than sorting.Here, a solution is presented that requires a linear number of comparisons for the following two cases: • The sets are hyperplanes in a d-dimensional projective geometry @math . In particular, the interesting case is @math , when the intersection of any two subsets is exactly one. • The sets are chosen randomly with probability @math for each element to be in a set. The random choices are mutually independent and the number of comparisons needed is linear with probability approaching 1 asymptotically. Unsupervised word embeddings provide rich linguistic and conceptual information about words. However, they may provide weak information about domain specific semantic relations for certain tasks such as semantic parsing of natural language queries, where such information about words can be valuable. To encode the prior knowledge about the semantic word relations, we present new method as follows: We extend the neural network based lexical word embedding objective function Mikolov, et.al. 2013 by incorporating the information about relationship between entities that we extract from knowledge bases. Our model can jointly learn lexical word representations from free text enriched by the relational word embeddings from relational data (e.g., Freebase) for each type of entity relations. We empirically show on the task of semantic tagging of natural language queries that our enriched embeddings can provide information about not only short-range syntactic dependencies but also long-range semantic dependencies between words. Using the enriched embeddings, we obtain an average of 2 improvement in F-score compared to the previous baselines. Many tasks in natural language processing, ranging from machine translation to question answering, can be reduced to the problem of matching two sentences or more generally two short texts. We propose a new approach to the problem, called Deep Match Tree (DeepMatch @math ), under a general setting. The approach consists of two components, 1) a mining algorithm to discover patterns for matching two short-texts, defined in the product space of dependency trees, and 2) a deep neural network for matching short texts using the mined patterns, as well as a learning algorithm to build the network having a sparse structure. We test our algorithm on the problem of matching a tweet and a response in social media, a hard matching problem proposed in [, 2013], and show that DeepMatch @math can outperform a number of competitor models including one without using dependency trees and one based on word-embedding, all with large margins A classic result of Johnson and Lindenstrauss asserts that any set of n points in d-dimensional Euclidean space can be embedded into k-dimensional Euclidean space---where k is logarithmic in n and independent of d--so that all pairwise distances are maintained within an arbitrarily small factor. All known constructions of such embeddings involve projecting the n points onto a spherically random k-dimensional hyperplane through the origin. We give two constructions of such embeddings with the property that all elements of the projection matrix belong in -1, 0, +1 . Such constructions are particularly well suited for database environments, as the computation of the embedding reduces to evaluating a single aggregate over k random partitions of the attributes. This paper proposes low-complexity algorithms for finding approximate second-order stationary points (SOSPs) of problems with smooth non-convex objective and linear constraints. While finding (approximate) SOSPs is computationally intractable, we first show that generic instances of the problem can be solved efficiently. More specifically, for a generic problem instance, certain strict complementarity (SC) condition holds for all Karush-Kuhn-Tucker (KKT) solutions (with probability one). The SC condition is then used to establish an equivalence relationship between two different notions of SOSPs, one of which is computationally easy to verify. Based on this particular notion of SOSP, we design an algorithm named the Successive Negative-curvature grAdient Projection (SNAP), which successively performs either conventional gradient projection or some negative curvature based projection steps to find SOSPs. SNAP and its first-order extension SNAP @math , require @math iterations to compute an @math -SOSP, and their per-iteration computational complexities are polynomial in the number of constraints and problem dimension. To our knowledge, this is the first time that first-order algorithms with polynomial per-iteration complexity and global sublinear rate have been designed to find SOSPs of the important class of non-convex problems with linear constraints.