# Expected Signature Kernels for Rough Paths

arXiv: https://arxiv.org/abs/2509.07893 · HTML: https://mlsanity.com/paper/2509.07893/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 35 references that are in the map.
- First posted on arXiv 9 Sep 2025.

## Builds on (most cited first, 7 of 7)

- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [Conditional Sig-Wasserstein GANs for Time Series Generation](https://mlsanity.com/paper/2006.05421/) (2020)
- [Approximate Bayesian Computation with Path Signatures](https://mlsanity.com/paper/2106.12555/) (2021)
- [Signature Maximum Mean Discrepancy Two-Sample Statistical Tests](https://mlsanity.com/paper/2506.01718/) (2025)

## Built on by (most cited first, 1 of 1)

- [Local regression on path spaces with signature metrics](https://mlsanity.com/paper/2510.16728/) (2025)

## Abstract (arXiv)

The expected signature kernel is a similarity measure of probability laws on path space and enters directly into signature maximum mean discrepancies. Building on [P. K. Friz, P. P. Hager, and N. Tapia, Forum Math. Sigma, 10 (2022), e42] and [M. Lemercier, T. Lyons, and C. Salvi, SIAM J. Numer. Anal., 64 (2026), pp. 1095-1125], we characterize the expected signature kernel of group-valued inhomogeneous L'evy processes, aka L'evy rough paths, as the scalar component of the solution to an extended Goursat-type system. We give an original description, first in the continuous case where the finite-dimensional system is exact. In the presence of jumps, this system is inherently infinite-dimensional; we prove its well-posedness and derive finite-dimensional truncations with explicit error control. This permits direct evaluation of the reference-dependent terms in one-sample signature-MMD tests without an auxiliary reference sample. Finally, for conditionally Gaussian martingales, conditioning the usual unbiased empirical squared signature-MMD on the sampled quadratic variations yields an unbiased statistic for the same quantity with no larger variance. We derive computable finite-grid versions and illustrate the resulting variance reduction and gain in power.