Learning First-Order Logic Embeddings via Matrix Factorization

Abstract

Many complex reasoning tasks in Artificial Intelligence (including relation extraction, knowledge base completion, and information integration) can be formulated as inference problems using a probabilistic first-order logic. However, due to the discrete nature of logical facts and predicates, it is challenging to generalize symbolic representations and represent first-order logic formulas in probabilistic relational models. In this work, we take a rather radical approach: we aim at learning continuous low-dimensional embeddings for first-order logic from scratch. In particular, we first consider a structural gradient based structure learning approach to generate plausible inference formulas from facts; then, we build grounded proof graphs using background facts, training examples, and these inference formulas. To learn embeddings for formulas, we map the training examples into the rows of a binary matrix, and inference formulas into the columns. Using a scalable matrix factorization approach, we then learn the latent continuous representations of examples and logical formulas via a low-rank approximation method. In experiments, we demonstrate the effectiveness of reasoning with first-order logic embeddings by comparing with several state-of-the-art baselines on two datasets in the task of knowledge base completion.