Reservoir kernels and Volterra series
This provides a new kernel method for sequential data analysis, applicable to domains like finance, but it appears incremental as it builds on existing Volterra series and reservoir concepts.
The authors tackled the problem of approximating causal, time-invariant filters with fading memory by constructing a universal kernel called the Volterra reservoir kernel, which is built from a state-space representation of Volterra series and demonstrated empirical performance in a financial asset returns task.
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, and it is hence called the Volterra reservoir kernel. Even though the state-space 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. The empirical performance of the Volterra reservoir kernel is showcased and compared to other standard static and sequential kernels in a multidimensional and highly nonlinear learning task for the conditional covariances of financial asset returns.