The signature is a non-commutative exponential that appeared in the foundational work of K-T Chen in the 1950s. It is also a fundamental object in the theory of rough paths (Lyons, 1998). More recently, it has been proposed, and used, as part of a practical methodology to give a way of summarising multimodal, possibly irregularly sampled, time-ordered data in a way that is insensitive to its parameterisation. A key property underpinning this approach is the ability of linear functionals of the signature to approximate arbitrarily any compactly supported and continuous function on (unparameterised) path space. We present some new results on the properties of a selection of topologies on the space of unparameterised paths. Relatedly, we review some recent innovations in the theory of the signature kernel by introducing and analysing the properties of a family of so-called weighted signature kernels
The talk will draw on material from two recent papers; one is joint work with William F. Turner, the other is a joint work with Terry Lyons and Xingcheng Xu.