Knowledge graphs are structured representations of real world facts. However, they typically contain only a small subset of all possible facts. Link prediction is a task of inferring missing facts based on existing ones. We propose TuckER, a relatively simple but powerful linear model based on Tucker decomposition of the binary tensor representation of knowledge graph triples. TuckER outperforms all previous state-of-the-art models across standard link prediction datasets. We prove that TuckER is a fully expressive model, deriving the bound on its entity and relation embedding dimensionality for full expressiveness which is several orders of magnitude smaller than the bound of previous state-of-the-art models ComplEx and SimplE. We further show that several previously introduced linear models can be viewed as special cases of TuckER.
RESCAL An early linear model, RESCAL @cite_18 , optimizes a scoring function containing a bilinear product between vector embeddings for each subject and object entity and a full rank matrix for each relation. Although a very expressive and powerful model, RESCAL is prone to overfitting due to its large number of parameters, which increases quadratically in the embedding dimension with the number of relations in a knowledge graph.
@cite_18
2951077644 2101043704 2024051019 1969415786
We consider learning representations of entities and relations in KBs using the neural-embedding approach. We show that most existing models, including NTN (, 2013) and TransE (, 2013b), can be generalized under a unified learning framework, where entities are low-dimensional vectors learned from a neural network and relations are bilinear and or linear mapping functions. Under this framework, we compare a variety of embedding models on the link prediction task. We show that a simple bilinear formulation achieves new state-of-the-art results for the task (achieving a top-10 accuracy of 73.2 vs. 54.7 by TransE on Freebase). Furthermore, we introduce a novel approach that utilizes the learned relation embeddings to mine logical rules such as "BornInCity(a,b) and CityInCountry(b,c) => Nationality(a,c)". We find that embeddings learned from the bilinear objective are particularly good at capturing relational semantics and that the composition of relations is characterized by matrix multiplication. More interestingly, we demonstrate that our embedding-based rule extraction approach successfully outperforms a state-of-the-art confidence-based rule mining approach in mining Horn rules that involve compositional reasoning. We design a new distribution over poly(r e-1) x n matrices S so that for any fixed n x d matrix A of rank r, with probability at least 9 10, SAx2 = (1 pm e)Ax2 simultaneously for all x ∈ Rd. Such a matrix S is called a subspace embedding. Furthermore, SA can be computed in O(nnz(A)) + O(r2e-2) time, where nnz(A) is the number of non-zero entries of A. This improves over all previous subspace embeddings, which required at least Ω(nd log d) time to achieve this property. We call our matrices S sparse embedding matrices. Using our sparse embedding matrices, we obtain the fastest known algorithms for overconstrained least-squares regression, low-rank approximation, approximating all leverage scores, and lp-regression: to output an x' for which Ax'-b2 ≤ (1+e)minx Ax-b2 for an n x d matrix A and an n x 1 column vector b, we obtain an algorithm running in O(nnz(A)) + O(d3e-2) time, and another in O(nnz(A)log(1 e)) + O(d3log(1 e)) time. (Here O(f) = f ⋅ logO(1)(f).) to obtain a decomposition of an n x n matrix A into a product of an n x k matrix L, a k x k diagonal matrix D, and a n x k matrix W, for which F A - L D W ≤ (1+e)F A-Ak , where Ak is the best rank-k approximation, our algorithm runs in O(nnz(A)) + O(nk2 e-4log n + k3e-5log2n) time. to output an approximation to all leverage scores of an n x d input matrix A simultaneously, with constant relative error, our algorithms run in O(nnz(A) log n) + O(r3) time. to output an x' for which Ax'-bp ≤ (1+e)minx Ax-bp for an n x d matrix A and an n x 1 column vector b, we obtain an algorithm running in O(nnz(A) log n) + poly(r e-1) time, for any constant 1 ≤ p In this paper, we propose a rank minimization method to fuse the predicted confidence scores of multiple models, each of which is obtained based on a certain kind of feature. Specifically, we convert each confidence score vector obtained from one model into a pairwise relationship matrix, in which each entry characterizes the comparative relationship of scores of two test samples. Our hypothesis is that the relative score relations are consistent among component models up to certain sparse deviations, despite the large variations that may exist in the absolute values of the raw scores. Then we formulate the score fusion problem as seeking a shared rank-2 pairwise relationship matrix based on which each original score matrix from individual model can be decomposed into the common rank-2 matrix and sparse deviation errors. A robust score vector is then extracted to fit the recovered low rank score relation matrix. We formulate the problem as a nuclear norm and l 1 norm optimization objective function and employ the Augmented Lagrange Multiplier (ALM) method for the optimization. Our method is isotonic (i.e., scale invariant) to the numeric scales of the scores originated from different models. We experimentally show that the proposed method achieves significant performance gains on various tasks including object categorization and video event detection. Variable selection in the linear regression model takes many apparent faces from both frequentist and Bayesian standpoints. In this paper we introduce a variable selection method referred to as a rescaled spike and slab model. We study the importance of prior hierarchical specifications and draw connections to frequentist generalized ridge regression estimation. Specifically, we study the usefulness of continuous bimodal priors to model hypervariance parameters, and the effect scaling has on the posterior mean through its relationship to penalization. Several model selection strategies, some frequentist and some Bayesian in nature, are developed and studied theoretically. We demonstrate the importance of selective shrinkage for effective variable selection in terms of risk misclassification, and show this is achieved using the posterior from a rescaled spike and slab model. We also show how to verify a procedure's ability to reduce model uncertainty in finite samples using a specialized forward selection strategy. Using this tool, we illustrate the effectiveness of rescaled spike and slab models in reducing model uncertainty.