  We propose a modular framework for multi-relational learning via tensor
decomposition. In our learning setting, the training data contains multiple
types of relationships among a set of objects, which we represent by a sparse
three-mode tensor. The goal is to predict the values of the missing entries. To
do so, we model each relationship as a function of a linear combination of
latent factors. We learn this latent representation by computing a low-rank
tensor decomposition, using quasi-Newton optimization of a weighted objective
function. Sparsity in the observed data is captured by the weighted objective,
leading to improved accuracy when training data is limited. Exploiting sparsity
also improves efficiency, potentially up to an order of magnitude over
unweighted approaches. In addition, our framework accommodates arbitrary
combinations of smooth, task-specific loss functions, making it better suited
for learning different types of relations. For the typical cases of real-valued
functions and binary relations, we propose several loss functions and derive
the associated parameter gradients. We evaluate our method on synthetic and
real data, showing significant improvements in both accuracy and scalability
over related factorization techniques.
