A High Order Solver for Signature Kernels
Maud Lemercier Terry Lyons
Signature kernels are at the core of several machine learning algorithms for
analysing multivariate time series. The kernel of two bounded variation paths
(such as piecewise linear interpolations of time series data) is typically
computed by solving a Goursat problem for a hyperbolic partial differential
equation (PDE) in two independent time variables. However, this approach
becomes considerably less practical for highly oscillatory input paths, as they
have to be resolved at a fine enough scale to accurately recover their
signature kernel, resulting in significant time and memory complexities. To
mitigate this issue, we first show that the signature kernel of a broader class
of paths, known as \emph{smooth rough paths}, also satisfies a PDE, albeit in
the form of a system of coupled equations. We then use this result to introduce
new algorithms for the numerical approximation of signature kernels. As bounded
variation paths (and more generally geometric $p$-rough paths) can be
approximated by piecewise smooth rough paths, one can replace the PDE with
rapidly varying coefficients in the original Goursat problem by an explicit
system of coupled equations with piecewise constant coefficients derived from
the first few iterated integrals of the original input paths. While this
approach requires solving more equations, they do not require looking back at
the complex and fine structure of the initial paths, which significantly
reduces the computational complexity associated with the analysis of highly
oscillatory time series.