{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/general-signature-kernels","title":"General Signature Kernels","arxiv_id":"2107.00447","date":"2021-07-01","proceeding":null,"authors":["Thomas Cass","Terry Lyons","Xingcheng Xu"],"abstract":"Suppose that $\\gamma$ and $\\sigma$ are two continuous bounded variation paths which take values in a finite-dimensional inner product space $V$. Recent papers have introduced the truncated and the untruncated signature kernel of $\\gamma$ and $\\sigma$, and showed how these concepts can be used in classification and prediction tasks involving multivariate time series. In this paper, we introduce general signature kernels and show how they can be interpreted, in many examples, as an average of PDE solutions, and hence how they can be computed using suitable quadrature rules. 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 hyperbolic development map, the latter of which has been a broadly useful tool in the analysis of the signature. As an application we evaluate the use of different general signature kernels as the basis for non-parametric goodness-of-fit tests to Wiener measure on path space.","url_abs":"https://arxiv.org/abs/2107.00447v1","url_pdf":"https://arxiv.org/pdf/2107.00447v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"general-signature-kernels","repo_url":"https://github.com/xingchengxu/WSigKernel","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2107.00447","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2107.00447"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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