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.
While 2-way models were shown to have very good performances on some KB datasets, they have limited expressiveness and they can fail dramatically on harder datasets. In contrast, 3-way models perform some form of low-rank tensor factorization, and in that respect can have extremely high expressiveness depending on the rank constraints. In the context of link prediction for multi-relational data, @cite_23 follows natural modeling assumptions. Similarly to , learns one low-dimensional embedding for each entity. However, relationships are represented as a bilinear operator in the embedding space, i.e. each relationship corresponds to a matrix. The training objective of is the Frobenius norm between the original data tensor and its low-rank reconstruction, whereas uses the margin ranking criterion of . Another related 3-way model is @cite_14 . The parameterization of is a constrained version of , and also uses a ranking criterion as training objective.
@cite_14 @cite_23
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There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher that parallels the Eckart-Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank- @math approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders, and ranks, regardless of the choice of norm (or even Bregman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank-3 tensor has an optimal rank-2 approximation. The notable exceptions to this misbehavior are rank-1 tensors and order-2 tensors (i.e., matrices). In a more positive spirit, we propose a natural way of overcoming the ill-posedness of the low-rank approximation problem, by using weak solutions when true solutions do not exist. For this to work, it is necessary to characterize the set of weak solutions, and we do this in the case of rank 2, order 3 (in arbitrary dimensions). In our work we emphasize the importance of closely studying concrete low-dimensional examples as a first step toward more general results. To this end, we present a detailed analysis of equivalence classes of @math tensors, and we develop methods for extending results upward to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory b In this paper, we exploit the advantages of tensorial representations and propose several tensor learning models for regression. The model is based on the canonical parallel-factor decomposition of tensors of multiple modes and allows the simultaneous projections of an input tensor to more than one direction along each mode. Two empirical risk functions are studied, namely, the square loss and e-insensitive loss functions. The former leads to higher rank tensor ridge regression (TRR), and the latter leads to higher rank support tensor regression (STR), both formulated using the Frobenius norm for regularization. We also use the group-sparsity norm for regularization, favoring in that way the low rank decomposition of the tensorial weight. In that way, we achieve the automatic selection of the rank during the learning process and obtain the optimal-rank TRR and STR. Experiments conducted for the problems of head-pose, human-age, and 3-D body-pose estimations using real data from publicly available databases, verified not only the superiority of tensors over their vector counterparts but also the efficiency of the proposed algorithms.