Core Concepts
The signature kernel of two smooth rough paths solves an augmented system of linear partial differential equations, which can be efficiently approximated by replacing the input paths with piecewise log-linear paths.
Abstract
The article presents a new approach for efficiently computing signature kernels, which are at the core of several machine learning algorithms for analyzing multivariate time series data.
Key highlights:
The signature kernel of two smooth rough paths X and Y solves an augmented system of linear PDEs, which generalizes the original Goursat problem for the signature kernel of two bounded variation paths.
The additional variables in the PDE system are the initial terms of ℓ(S(X))(S(Y)) and ℓ(S(Y))(S(X)), which capture higher-order information about the input paths.
The PDE system can be decomposed into two parts: one that determines the signature kernel, and another that computes the additional state variables. This allows efficient computation of the kernel without the need to resolve the fine structure of the input paths.
The authors leverage this result to derive high-order numerical schemes for computing signature kernels of arbitrary rough paths. The key idea is to approximate the input paths by piecewise log-linear paths, which are a natural extension of classical piecewise linear approximations.
The piecewise log-linear approximation replaces 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 log-signatures of the original input paths.
The proposed approach significantly reduces the computational complexity associated with the analysis of highly oscillatory time series data, as it does not require looking back at the complex and fine structure of the initial paths.