  In recent years, hyper-complex deep networks (such as complex-valued and
quaternion-valued neural networks) have received a renewed interest in the
literature. They find applications in multiple fields, ranging from image
reconstruction to 3D audio processing. Similar to their real-valued
counterparts, quaternion neural networks (QVNNs) require custom regularization
strategies to avoid overfitting. In addition, for many real-world applications
and embedded implementations, there is the need of designing sufficiently
compact networks, with few weights and neurons. However, the problem of
regularizing and/or sparsifying QVNNs has not been properly addressed in the
literature as of now. In this paper, we show how to address both problems by
designing targeted regularization strategies, which are able to minimize the
number of connections and neurons of the network during training. To this end,
we investigate two extensions of l1 and structured regularization to the
quaternion domain. In our experimental evaluation, we show that these tailored
strategies significantly outperform classical (real-valued) regularization
approaches, resulting in small networks especially suitable for low-power and
real-time applications.
