Projected Canonical Decomposition for Knowledge Base Completion

The leading approaches to tensor completion and link prediction are based on the canonical polyadic (CP) decomposition of tensors. While these approaches were originally motivated by low rank approximations, the best performances are usually obtained for ranks as high as permitted by computation constraints. For large scale factorization problems where the factor dimensions have to be kept small, the performances of these approaches tend to drop drastically. The other main tensor factorization model, Tucker decomposition, is more flexible than CP for fixed factor dimensions, so we expect Tucker-based approaches to yield better performance under strong constraints on the number of parameters. However, as we show in this paper through experiments on standard benchmarks of link prediction in knowledge bases, ComplEx, a variant of CP, achieves similar performances to recent approaches based on Tucker decomposition on all operating points in terms of number of parameters. In a control experiment, we show that one problem in the practical application of Tucker decomposition to large-scale tensor completion comes from the adaptive optimization algorithms based on diagonal rescaling, such as Adagrad. We present a new algorithm for a constrained version of Tucker which implicitly applies Adagrad to a CP-based model with an additional projection of the embeddings onto a fixed lower dimensional subspace. The resulting Tucker-style extension of ComplEx obtains similar best performances as ComplEx, with substantial gains on some datasets under constraints on the number of parameters.