Parameterization of Hypercomplex Multiplications

Recent works have demonstrated reasonable success of representation learning in hypercomplex space. Specifically, the Hamilton product (4D hypercomplex multiplication) enables learning effective representations while saving up to 75% parameters. However, one key caveat is that hypercomplex space only exists at very few predefined dimensions. This restricts the flexibility of models that leverage hypercomplex multiplications. To this end, we propose parameterizing hypercomplex multiplications, allowing models to learn multiplication rules from data regardless of whether such rules are predefined. As a result, our method not only subsumes the Hamilton product, but also learns to operate on any arbitrary nD hypercomplex space, providing more architectural flexibility. Experiments of applications to LSTM and Transformer on natural language inference, machine translation, text style transfer, and subject verb agreement demonstrate architectural flexibility and effectiveness of the proposed approach.