The Signature Kernel

The signature kernel is a positive definite kernel for sequential data. It inherits theoretical guarantees from stochastic analysis, has efficient algorithms for computation, and shows strong empirical performance. In this chapter, we provide an introduction to the signature kernel by highlighting the analytic connection between the path signature and ordinary monomials. In particular, both the classical monomials and the path signature are universal and characteristic on compact domains: they can approximate functions, and characterize probability measures. However, issues arise in practice due to unbounded domains, computational complexity, and non-robustness. We show that these issues can be avoided via kernelization and robustification. To address the computational complexity, we provide an overview of the kernel trick: algorithms to efficiently compute the signature kernel which avoid direct computations of the signature. Furthermore, the signature kernel is highly flexible, and provides a canonical way to turn a given kernel on any domain into a kernel for sequences in that domain, while retaining its theoretical and computational properties. Finally, we survey applications and recent developments of the signature kernel.