Abelian Neural Networks
This work addresses the challenge of incorporating algebraic structures into neural networks for tasks like word analogy, offering a novel approach with size-generalization, though it is incremental in its application to specific domains.
The authors tackled the problem of modeling binary operations with algebraic properties, constructing neural networks for Abelian group and semigroup operations that achieve universal approximation and size-generalization for multiset inputs. They demonstrated improved performance in a word analogy task over word2vec and a naive method, with concrete gains in accuracy.
We study the problem of modeling a binary operation that satisfies some algebraic requirements. We first construct a neural network architecture for Abelian group operations and derive a universal approximation property. Then, we extend it to Abelian semigroup operations using the characterization of associative symmetric polynomials. Both models take advantage of the analytic invertibility of invertible neural networks. For each case, by repeating the binary operations, we can represent a function for multiset input thanks to the algebraic structure. Naturally, our multiset architecture has size-generalization ability, which has not been obtained in existing methods. Further, we present modeling the Abelian group operation itself is useful in a word analogy task. We train our models over fixed word embeddings and demonstrate improved performance over the original word2vec and another naive learning method.