Weighted signature kernels

Abstract

Suppose that γ and σ are two continuous bounded variation paths, which take values in a finite-dimensional inner product space V. The recent papers (J. Mach. Learn. Res. 20 (2019) 1–45) and (SIAM J. Math. Data Sci. 3 (2021) 873–899), respectively, introduced the truncated and the untruncated signature kernel of γ and σ, and showed how these concepts can be used in classification and prediction tasks involving multivariate time series. In this paper, we introduce signature kernels ${\mathit{K}}_{\mathit{\varphi }}^{\mathit{\gamma },\mathit{\sigma }}$ indexed by a weight function ϕ, which generalise the ordinary signature kernel. We show how ${\mathit{K}}_{\mathit{\varphi }}^{\mathit{\gamma },\mathit{\sigma }}$ can be interpreted in many examples as an average of PDE solutions, and thus we show how it can be estimated computationally using suitable quadrature formulae. We extend this analysis to derive closed-form formulae for expressions involving the expected (Stratonovich) signature of Brownian motion. In doing so, we articulate a novel connection between signature kernels and the notion of the hyperbolic development of a path, which has been a broadly useful tool in the recent analysis of the signature; see, for example, (Ann. of Math. (2) 171 (2010) 109–167; J. Funct. Anal. 272 (2017) 2933–2955) and (Trans. Amer. Math. Soc. 372 (2019) 585–614). As applications, we evaluate the use of different general signature kernels as a basis for nonparametric goodness-of-fit tests to Wiener measure on path space.