  Hypercomplex neural networks have proven to reduce the overall number of
parameters while ensuring valuable performance by leveraging the properties of
Clifford algebras. Recently, hypercomplex linear layers have been further
improved by involving efficient parameterized Kronecker products. In this
paper, we define the parameterization of hypercomplex convolutional layers and
introduce the family of parameterized hypercomplex neural networks (PHNNs) that
are lightweight and efficient large-scale models. Our method grasps the
convolution rules and the filter organization directly from data without
requiring a rigidly predefined domain structure to follow. PHNNs are flexible
to operate in any user-defined or tuned domain, from 1D to $n$D regardless of
whether the algebra rules are preset. Such a malleability allows processing
multidimensional inputs in their natural domain without annexing further
dimensions, as done, instead, in quaternion neural networks for 3D inputs like
color images. As a result, the proposed family of PHNNs operates with $1/n$
free parameters as regards its analog in the real domain. We demonstrate the
versatility of this approach to multiple domains of application by performing
experiments on various image datasets as well as audio datasets in which our
method outperforms real and quaternion-valued counterparts. Full code is
available at: https://github.com/eleGAN23/HyperNets.
