The Signature Kernel Is the Solution of a Goursat PDE

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 the potential to handle irregularly sampled, multivariate time series. In [F. J. Király and H. Oberhauser, J. Mach. Learn. Res., 20 (2019), 31] 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-art hyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from Király and Oberhauser, 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 problem. Finally, we empirically demonstrate the effectiveness of this PDE kernel as a machine learning tool in various data science applications dealing with sequential data. We make the library \tt sigkernel publicly available at https://github.com/crispitagorico/sigkernel.