  This is the first paper in a sequence of studies in which we introduce a new
type of neural networks (NNs) -- wavelet-based neural networks (WBNNs) -- and
study their properties and potential for applications. We begin this study with
a comparison to the currently existing type of wavelet neural networks (WNNs)
and show that WBNNs vastly outperform WNNs. One reason for the vast superiority
of WBNNs is their advanced hierarchical tree structure based on biorthonormal
multiresolution analysis (MRA). Another reason for this is the implementation
of our new idea to incorporate the wavelet tree depth into the neural width of
the NN. The separation of the roles of wavelet depth and neural depth provides
a conceptually and algorithmically simple but highly efficient methodology for
sharp increase in functionality of swarm and deep WBNNs and rapid acceleration
of the machine learning process.
