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The signature kernel is the solution of a Goursat PDE

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Title: The signature kernel is the solution of a Goursat PDE
Authors: Salvi, C
Cass, T
Foster, J
Lyons, T
Yang, W
Item Type: Journal Article
Abstract: Recently, there has been an increased interest in the development of kernel methods for learning with sequential data. The signature kernel is a learning tool with potential to handle irregularly sampled, multivariate time series. In [1] the authors introduced a kernel trick for the truncated version of this kernel avoiding the exponential complexity that would have been involved in a direct computation. Here we show that for continuously differentiable paths, the signature kernel solves a hyperbolic PDE and recognize the connection with a well known class of differential equations known in the literature as Goursat problems. This Goursat PDE only depends on the increments of the input sequences, does not require the explicit computation of signatures and can be solved efficiently using state-of-the-art hyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from [1], but with the advantage that the PDE numerical scheme is well suited for GPU parallelization, which effectively reduces the complexity by a full order of magnitude in the length of the input sequences. In addition, we extend the previous analysis to the space of geometric rough paths and establish, using classical results from rough path theory, that the rough version of the signature kernel solves a rough integral equation analogous to the aforementioned Goursat problem. Finally, we empirically demonstrate the effectiveness of this PDE kernel as a machine learning tool in various data science applications dealing with sequential data. We release the library sigkernel publicly available at https://github.com/crispitagorico/sigkernel
Issue Date: 9-Sep-2021
Date of Acceptance: 4-Jun-2021
URI: http://hdl.handle.net/10044/1/89404
DOI: 10.1137/20M1366794
ISSN: 2577-0187
Publisher: Society for Industrial and Applied Mathematics
Start Page: 873
End Page: 899
Journal / Book Title: SIAM Journal on Mathematics of Data Science
Volume: 3
Issue: 3
Copyright Statement: © 2021, Society for Industrial and Applied Mathematics.
Sponsor/Funder: Engineering & Physical Science Research Council (E
Funder's Grant Number: BKR01300
Keywords: Science & Technology
Physical Sciences
Mathematics, Applied
path signature
Goursat PDE
sequential data
geometric rough path
rough integration
60L10, 60L20
Publication Status: Published
Online Publication Date: 2021-09-09
Appears in Collections:Financial Mathematics
Faculty of Natural Sciences