  Recently, there has been an increased interest in the development of kernel
methods for learning with sequential data. The signature kernel is a learning
tool with potential to handle irregularly sampled, multivariate time series. In
"Kernels for sequentially ordered data" the authors introduced a kernel trick
for the truncated version of this kernel avoiding the exponential complexity
that would have been involved in a direct computation. Here we show that for
continuously differentiable paths, the signature kernel solves a hyperbolic PDE
and recognize the connection with a well known class of differential equations
known in the literature as Goursat problems. This Goursat PDE only depends on
the increments of the input sequences, does not require the explicit
computation of signatures and can be solved efficiently using
state-of-the-arthyperbolic PDE numerical solvers, giving a kernel trick for the
untruncated signature kernel, with the same raw complexity as the method from
"Kernels for sequentially ordered data", but with the advantage that the PDE
numerical scheme is well suited for GPU parallelization, which effectively
reduces the complexity by a full order of magnitude in the length of the input
sequences. In addition, we extend the previous analysis to the space of
geometric rough paths and establish, using classical results from rough path
theory, that the rough version of the signature kernel solves a rough integral
equation analogous to the aforementioned Goursat PDE. Finally, we empirically
demonstrate the effectiveness of our PDE kernel as a machine learning tool in
various machine learning applications dealing with sequential data. We release
the library sigkernel publicly available at
https://github.com/crispitagorico/sigkernel.
