Reservoir kernels and Volterra series

A universal kernel is constructed whose sections
approximate any causal and time-invariant filter in the fading
memory category with inputs and outputs in a finite-dimensional
Euclidean space. This kernel is built using the reservoir functional
associated with a state-space representation of the Volterra series
expansion available for any analytic fading memory filter. It is
hence called the Volterra reservoir kernel. Even though the statespace
representation and the corresponding reservoir feature
map are defined on an infinite-dimensional tensor algebra space,
the kernel map is characterized by explicit recursions that are
readily computable for specific data sets when employed in
estimation problems using the representer theorem. We showcase
the performance of the Volterra reservoir kernel in a popular
data science application in relation to bitcoin price prediction.